It is common fare among secularists and opponents of Christianity in general to cite alleged similarities between early Christianity and Mithraism as an argument that the Church adopted many of its beliefs and practices from one or more pagan mystery religions. Certain superficial similarities are brought forward: an alleged Mithraic "baptism" that was similar to Christian baptism, the ritual consumption of the god, an elite hierarchical organization, certain parallels between the lives of Christ and Mithra, etc. The purpose for dwelling on these alleged similarities is usually to deny the supernatural origin of Christianity and its doctrines, proposing as an alternative that Christianity was a synthesis of various preexisting pagan religions. This is commonly asserted by the "Christianity is pagan" crowd.
Influence of Franz Cumont
These theories all trace their origin to a single source: the work of the Belgian archaeologist Franz Cumont (1868-1947) whose 1903 book Texts and Illustrated Monuments Relating to the Mysteries of Mithra entirely dominates the study of the cult of Mithra and of ancient Roman religion in general. Any contemporary book that discusses Mithraism most likely goes to Cumont as its sole source. And while most scholarship on Roman Mithraism goes back to Cumont, most of Cumont's information came from the excavation of a single site, that of the Mithraeum at Dura-Europos in Syria which was excavated in the early 20th century. Much of the work Cumont subsequently published was plagued by unwarranted assumptions, which have unfortunately bled into the study of Mithraism in general.
For example, two figures depicted in the Mithraeum were assumed by Cumont to be the Persian personalities of Zoroaster and Ostanes, although this was a pure assumption and, based on what has been found at other Mithraeums, could very well have been local elders in the Mithraic community. Here we see a predisposition to assuming continuity with the Iranian cult leading to premature designations on the part of Cumont. He also errantly states that Mithraism was the leading religious cult of the third century, justifying this by incredibly vague references to "the misery of the times" and the desire of Romans for the security offered by the Mithraic rites, which accounted for its popularity. In fact, according to epigraphic evidence, the most popular cults in the 3rd century Roman Empire were those of Jupiter, then Mercury and Hercules. The cult of Mithras comes in a distant 9th place, below gods such as Fortuna, Silvanus and Liber, and even then only when combined with Sol, with whom he is commonly identified. If Mithraism was the dominant cult of the 3rd century, it is odd that more Mithraic inscriptions have not survived, since the subterranean Mithraeums lend themselves naturally to archaeological preservation. 
Cumont is also responsible for some of the most commonly repeated falsehoods about Mithraism. For example, the assertions that Mithraism was hierarchically structured, that it was a soldier's religion, that they believed in moral dualism and had notions of sin and atonement. All of these assumptions come from Cumont and are all believed to be "wholly unsubstantiated" by a growing cadre of eminent historians.  The most commonly repeated assertion about Mithraism, that it excluded women, is also without foundation, especially in light of the fact that several Mithraic inscriptions exist which were carved or commissioned by women.
As a further example of the kind of free speculation that has unfortunately plagued Mithraic studies, we can cite the work of the scholar I.F. Legge, who opined (1912) that Mithraic lion-headed figure common in Mithraeums was the Zoroastrian god Ahriman. This assumption was based on nothing other than the fact that the name Arimanius has been discovered in Roman Mithraic epigraphy - nevermind the fact that the name never occurs in any context that could possibly identify Arimanius with Ahriman.  Again, there is no evidence suggesting a Persian connection other than the historian's prejudice that a Persian connection exists. It is a classic case of petitio principii.
What does all this mean? Essentially it indicates that the narrative Cumont proposed of a very ascendant and prominent Mithraism competing with an equally ascendant Christianity, in which the latter finally triumphed by appropriating elements of the former, is not backed up by epigraphic evidence and is rendered more questionable by some of the faulty assumptions of Cumont. Cumont's work was even rebuked at the First International Congress of Mithraic Studies in 1971, where a panel of Mithraic experts publicly criticized Cumont's assertion that Roman Mithra was the same as the Iranian deity. So, when examining the question of Roman Mithraism, we need to go beyond Cumont and the assumptions inherent in his work.
Iranian or Roman Origin?
The gist of the critique leveled against Cumont is that there is insufficient evidence that the Roman cult of Mithras related in any way to the earlier Persian cult. Cumont's "continuity theory" is central to the assertions of syncretist Christian borrowings from Mithraism.
According to Cumont, the Roman god Mithras is believed to be based on a Persian predecessor, who in Old Persian was called Miça, and Mihr in Parthian. This Miça is a Zoroastrian angelic being who is associated with the dawn and with cattle and is one of the three judges of men at the end of life, not unlike the role assigned to Rhadamanthus in Greek mythology. Interestingly enough, while worship of Miça is attested going back to pre-Hellenistic times, the original Iranian Miça bears little resemblance to the Roman Mithras. Cumont believed that the Persian worship of Miça was transferred from the east to the west through the intermediacy of Syria, but contemporary research suggests the Roman cult of Mithras was an original creation of the west and has no meaningful connection with the ancient Persian religion.
For example, the earliest reference to Mithraism in the Roman Empire comes from a single monument in the province of Cilicia in Asia Minor in 77 AD. Some Phrygian coins from the reign of Domitian bearing the image of Mithras are the next reference. A monument appears on the Danube around the year 100. After the reign of Trajan, the monuments and inscriptions begin to multiply. Thus, as far as the actual record goes, Roman Mithraism appears to have originated in the Cilician-Phrygian area of Asia Minor in the mid 1st century AD. This is consonant with depictions of the Roman Mithras, who is shown wearing a Phrygian cap, pictured at right. 
The Roman historian Plutarch also assigns a Phrygian origin to Mithraism. Writing his Life of Pompey in the early 2nd century, Plutarch tangentially mentions the religious customs of the peoples of Cilicia and Phrygia and states that the Cilicians "celebrated there certain secret rites, among which those of Mithras continue to the present time, having been first instituted by them."  This is the first literary mention of Mithraism, and it was stated to be Cilician in origin. It was not until much later - the early third century - that the Romans began suggesting a Persian connection to Mithras, such as wee see in the works of Porphyry.
Roman historian Ramsay MacMullen opines that the Roman cult of Mithras may have originally come from the Danube provinces and been brought east:
"Mithraism's ties with the east amount to so little that they can be denied entirely: it was rather created "at a defined moment by some unknown religious genius," best, in the Danube provinces. So say some scholars. When it is found in Dura on the Euphrates in the third century, its presence there is rightly explained as secondary, brought by soldiers from Palmyra who had learned it in their service with legions from Europe." 
Another Roman Mithraic scholar, Manfred Clauss, also believes that Mithraism has a European-Roman origin dating from the late 1st century. 
Furthermore, if Roman Mithraism was an evolution of a much earlier Iranian precedent, we should see some intermediate forms of it in the east, say in Anatolia or Syria, especially the former since Mithraism is first attested in Cilicia. Iranian archaeologists and cultural anthropologists have yet to identify any intermediate form of Mithraism in the region. The Encyclopedia Iranica states that "archaeology has (as yet) unearthed no evidence in Anatolia for an intermediate form of Mithras-worship which is unambiguously the precursor of the Roman mystery cult."  The same article, when examining the lack of evidence for a pre-Roman Mithraism in Persia in context of the theories of Cumont, states that "the absence of data on any intermediary form of Mithraism...is remarkable " and agrees with MacMullen that Roman Mithraism may have come to Syria from the west, as opposed to a Persian transplant from the east. 
Another strong indication of the Roman origins of Mithraism concerns the most famous event in the Mithras mythology, the slaying of the bull. This killing of the bull by Mithras is the most common motif in Roman Mithraic art. The killing of the bull is supposed to be the central act of Mithras' life, the event that gives signification to the supposed sacred meal that Mithraic worshipers partook in. Yet, for the supposed centrality of this act in the Mithraic cult, the bull story does not appear at all in any Perisan sources. It appears to be a Roman invention. The bull killing is not extrinsic to Roman Mithraism; according to the Encyclopedia Iranica, it represented "the god’s principal act...the icon which represents it was so clearly the cult’s primary locus of meaning" . That being the case, if the Persian worship of Miça is essentially the same as the Roman worship of Mithras, how could this fundamental focus of the cult be lost in translation? It would be as if, in the spreading of Christianity from Palestine to Europe, the entire fact of the crucifixion was lost. How could this be considered the same religion?
The facts seem to suggest that Roman Mithraism is not derived from the Persian worship of Miça, but is in fact a religious cult indigenous to Europe and transferred back to the Middle East in the first century AD. It is true that later Romans believed Mithraism to be Persian, but the speculations and wild etymological assumptions of the ancient Romans do not always parallel actual history; in this case, they clearly do not.
Those favoring a syncretist origin for Christianity will also dwell on certain similarities between Mithraism and Christianity. A passage from Tertullian is frequently cited:
"Blush, you fellow-soldiers of his, henceforth not to be condemned even by him, but by some soldier of Mithras, who, at his initiation in the gloomy cavern, in the camp, it may well be said, of darkness, when at the sword's point a crown is presented to him, as though in mimicry of martyrdom, and thereupon put upon his head, is admonished to resist and cast it off, and, if you like, transfer it to his shoulder, saying that Mithras is his crown. And thenceforth he is never crowned; and he has that for a mark to show who he is, if anywhere he be subjected to trial in respect of his religion; and he is at once believed to be a soldier of Mithras if he throws the crown away— if he say that in his god he has his crown. Let us take note of the devices of the devil, who is wont to ape some of God's things with no other design than, by the faithfulness of his servants, to put us to shame, and to condemn us." 
The phrase about Mithraism aping Christianity is assumed to mean that there were great similarities between the two religions, such that Tertullian around 200 was touchy about the degree to which the two cults resembled one another. Nevermind that the similarities he is referring to here are unnamed other than the fact of Mithras worshipers regarding themselves as soldiers and refusing a crown just as the martyrs are soldiers of Christ and can be easily distinguished by their refusal to sacrifice. Nothing else is noted here.
Other similarities have been put forward: believers partook in a sacred feast where they ritually consumed the god, Mithras' virgin birth, a ceremony of baptism by which believers are cleansed and promised immortality by the savior-god, a hierarchical organization, etc. Most of these similarities are ephemeral at best; for one thing, very little is known about what Mithraists believed or how they behaved. Some points, like the existence of a hierarchy, are completely fabricated and based on assumptions by Cumont and others. The other points are equally questionable. When looking at these alleged similarities, we must remember that, as Manfred Clauss states, "we possess virtually no theological statements either by Mithraists themselves or by other writers." 
Another example is the word "Savior" applied to Mithras, which is considered a parallel to Christ. While it is true that many gods, especially Asclepius, are called "Savior", the terms "Savior" and "salvation" in the Roman Empire denoted only physical health and had no connotations of the afterlife of immortality. They always referred to matters of life on this earth and temporal prosperity; the same is true of many of the other "mystery religions", such as the cult of Zeus of Panamara. There is no evidence whatsoever that Mithraism promised personal immortality, or that any mystery religion did so, for that matter.  As far as we know, Christianity is the first religion to definitively promise personal immortality.
Then there is the much touted tauobolium, the supposed Mithraic "baptism" whereby an initiate is drenched by the blood of a bull sacrificed above him. This is patently false. The taurobolium was part of the worship of Cybele and Diana but was not part of Mithraic worship. There is no historic source, literary, archaeological, or epigraphic, which connects Mithraism with the taurobolium. The misconception arises from mixing up the Mithraic motif of the slaying of the bull with the actual rite of sacrifice carried out in cults to Cybele, although it also must be stressed that neither is there any proven connection between the rites of Cybele and those of Roman Mithras. This is just a case of misapplication and ignorance.
Mithras' alleged "virgin birth" is also based on a misunderstanding, for most Roman Mithraic texts have him emerging fully formed in the style of Athena from a rock. .
The Mithraists certainly had sacred feasts, but so did every other religious association in the Roman Empire, so this is hardly surprising. As for consuming the flesh of the god, this would be strange, since in the mythic scenes decorating the surviving Mithraeum, it is Mithras who kills and feasts on the bull, not Mithras who is killed and feasted upon. All religions of the Roman era, including Christianity, utilized sacred feasts. Manfred Clauss states that the alleged comparisons between the two religions and whether Christianity absorbed Mithraism is a canard:
"the entire discussion is largely unhistorical. To raise the issue of a competition between the two religions is to assume that Christians and Mithraists had the same aims. Such a view exaggerates the missionary zeal — itself a Christian idea — of the other mystery cults. None of them aimed to become the sole legitimate religion of the Roman empire, because they offered an entirely individual and personal salvation. The alternative ‘Mithras or Christ?’ is wrongly framed, because it postulates a competitive situation which, in the eyes of Mithraists, simply did not exist….We should not simply transpose Christian views and terms in this area onto other mystery cults. Most of the parallels between Mithraism and Christianity are part of the common currency of all mystery cults or can be traced back to common origins in the Graeco-oriental culture of the Hellenistic world. The similarities do not at all suggest mutual influence…" 
The fact is, we know nothing about Mithraic feasts, and any other alleged claims about Mithraic doctrine (belief in Mithras' resurrection, etc). The connections between Persia and Roman Mithraism, and between Christianity and Mithraism or Christ and Mithras, are entirely overblown.
The assertion that Christianity borrowed from some pre-Christian Persian religion via the Roman cult of Mithras does not stand up to historical scrutiny. Far from being a pre-Christian Persian cult, it appears that Mithraism actually appeared on the scene later than Christianity, and in Europe, not Persia. There is no real historical link between Roman Mithraism and the Persian worship of Miça; in fact, it is not unreasonable to think that to the degree Mithraism did look like Christianity (and there is no good evidence that it did), it may have in fact borrowed from the Church rather than the other way around, since the Church predated Mithraism, at least according to epigraphic evidence. The view put forth by Cumont and others that Christianity is a syncretic blend of Mithraism and other mystery religions is historically untenable. The condemnation of the view becomes stronger when we realize that many of these alleged "similarities" upon which the theory rests either do not exist or are the results of a profound misunderstanding of religion in the Roman Empire. Since the 1970's Cumont's views have come under increasing scrutiny, to the point that reputable scholars now say that any pre-Christian Persian connection can safely be, in the words of secular historian Ramsay MacMullen, "denied entirely".
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 Ramsay MacMullen, Paganism in the Roman Empire (New Haven, CT: Yale University Press, 1981), 6
 ibid., 203
 J. Duchesne-Guillemin, “Ahriman et le dieu suprême dans les mystères de Mithra,” Numen 2, 1955, pp. 190-5.
 ibid., 118
 Plutarch, Parallel Lives, "Life of Pompey", 24
 MacMullen, 119
 Manfred Clauss, The Roman Cult of Mithras, trans. R.L. Gordon (New York: Routledge, 2001)
 Tertullian, de Corona, 15
 Clauss, 21
 MacMullen, 57
 Clauss, 168-169