It bears stating that Roman religion was much more practical than we tend to view the concept of belief from modern examples. It wasn't focused on how fervent your faith was. Instead the point was in correctly performing the necessary prayers and rituals to curry favour with a specific deity. Above all, Roman religion was a tool of social order, which even the more skeptical Romans viewed as a vital role. As religion in the Empire was polytheistic, nothing prevented Mithraists from participating in the civic religion of Rome, or even multiple other mystery cults.
One key factor is that as a mystery religion, the Mysteries of Mithras, as it was referred to at the time (Mithraism being a modern term), didn't function in a proselytizing manner like we would be familiar with from most current religions. It was meant to be a secret religion, dominated by soldiers and what we would consider the lower middle class today (tradesmen, merchants, low level officials & bureaucrats), as well as large numbers of slaves and freemen. The aristocracy didn't really participate until late in the history of the cult, during the brief pagan revival mid-4th century CE.
That said, like many other mystery religions or "occult" organizations, from what we do know, Mithraism operated as almost a trade union or guild with ceremonial and ritual practices suited to its members. If we were drawing modern comparisons, I'd say that it would be more closely comparable to the Freemasons than any of the Abrahamic religions, though this is necessarily reductive.
Lacking higher-class social support, not actively "recruiting" new accolytes, and lacking dogmatic "belief" necessarily put Mithraism (and other mystery religions) at a disadvantage in the face of the ascendant Christian religion.
Mithraism was finally extinguished by the end of the 4th century as Christianity became powerful enough to forcefully supplant rivals.
Emperor Theodosius I instituted his "Theodosian decrees" in 389–392 which abolished the remaining vestiges of Roman religion. While it's impossible to finally date the extinction of Mithraism due to the lack of historical records, there is evidence that gives us a decent reference point, particularly a large number of coins discovered at the Mithraeum in Sarrebourg. They were scattered across the floor when the site was destroyed, likely due to being seen as "unclean" by the Christians, as they were offerings to "false gods". The newest coins found were from the reign of Theodosius, which ended in 395.
You'll also find that often churches were built on the former sites of Mithraic activities, as a spiritual and cultural sign that the old "pagan" ways had been supplanted.
The thoroughness with which early Christianity attacked/persecuted rivals is exemplified by the fact that to this day, we have no documented theology for Mithraism. What we do know comes from interpretation of the physical evidence and inscriptions, as well as what we can glean from casual references in the literature of the time.
Further reading:
Cosmic Mysteries of Mithras, by David Ulansey summarizes his book "The Origins of the Mithraic Mysteries" (Oxford University Press).
The Pagan Shadow of Christ? by Roger Beck.
Web archive of papers from the Electronic Journal of Mithraic Studies
The Sacrifices of Mithrais by Britt-Marie Näsström.
Tl;DR: Mithraism was completely gone by the 5th Century, as it had been exterminated/supplanted by Christianity.