Like any account of a movement that grew from a small core group to a massive entity, you can understand that the causes and history of development is a complex set of phenomena and can hardly be reduced to any ‘Factor X’.

Nor should you discount the Resurrection of Jesus as one of those factors. Not whether the event occurred or not, but it’s quite clear that the disciples of Jesus, both those from Galilee and those who joined the movement in its very earliest days (i.e. in the first few years after the execution) whole-heartedly believed that Jesus had been raised from the dead, and took this as both a vindication of Jesus’ status in relationship to God (that he was indeed the Messiah they had awaited) and a sign of the beginning of a messianic age (though exactly how it ushered in a new messianic age was something they were figuring out and is a point of debate today – what exactly did Jesus expect, what did he teach, what did the first generation of disciples believe about ‘end times’, etc..). In one sense, it is irrelevant for the broader history of Christianity whether a person named Jesus came back to life, it is only relevant that this was a core belief of his first generation of disciples. (It is a relevant question to ask how they came to that belief and what they understood by it, but that is another question)

The fact that all NT documents are written in Greek is itself a very telling fact. The movement ‘re-geared’ itself from Jewish peasantry towards cities, and towards Greek as the language of its spread. This factor, the clear missionary zeal of the first and second generation of believers, and the huge role of Paul in the early church, lead to rapid expansion typical of new religious groups. One should remember, though, that even when I say ‘rapid expansion’ we are talking maybe 3.4 percent a year (this figure is from Rodney Stark, who compares growth rates with rates for aggressively proselytising groups like the Jehovah’s Witnesses and the Mormons, see The Triumph of Christianity page 156.).

The genius of Paul is seen in the fact that he (a) is responsible for a huge proportion of the NT corpus, (b) appears to have spear-headed evangelistic efforts in major cities across the eastern half of he empire, traversing the most populous provinces and focussed on major cities and trade-centers, so that his distinct apostolate of ‘mission to the Gentiles’, and the subsequent history of Mediterranean based Christianity is shaped both historically and theologically by Paul. Of course, Christianity was not exclusively an Empire phenomenon, but I suspect an account of Christianity in the Parthian and Sasanian Empire is not what you wanted today. Wilken suggests that by 150AD you could find Christian groups ranging from 20-200 people across 40-50 major cities within the Empire (Wilken, The Christians as the Romans Saw Them 1984, p31).

This increases greatly over the next 2 centuries, but again not at a huge rate, so that even by the early 4th century most historians put the percentage of Christians as growing only to roughly 10% of the population. Persecution was sporadic, but real, and Christians represented a marginal social group that was seen as atheistic, immoral, and socially disruptive. That did not stop it continuing to grow, primarily through social networks, personal relationships, and one should not discount the practice of Christian charity which, while not without non-Christian parallels, definitely differed from and oft times exceeded the kind of care for others promoted by Hellenistic and Roman value systems.

As well known and written about, the conversion of Constantine at the Milvian bridge, however it is parsed, represents a decisive turning point. After this, Constantine begins to favour Christians, and with the Edict of Milan extends religious tolerance. By 350 more than half the Empire are probably Christian, and its social standing has changed remarkably. By Theodosius in 379, a large majority would be Christians, and Theodosius moves Christianity from de facto religion of the Empire to ‘Official Religion’. This paves the way for the Christianisation of Europe that occurs through the Early Middle Ages and the history of Christianity into later periods.

We don't have the data to really know with a lot of certainty, but one key factor was definitely that the Jesus cult was de-Judaized and exported to Gentiles. Decoupling Christianity from Jewish law made it much easier to evangelize and expanded it from a Jewish sect to a cosmopolitan one.

I would say (and I'm not alone) that the survival and success of Christianity owes more to Paul than to Jesus.

One aspect that hasn't been touched on is the cost of participating in the religions of the day, an argument made by the New Testament scholar Marcus Borg. Full participation in many religions, including Temple Judaism and the Roman religion, could be a rather expensive proposition as it involved procuring animal sacrifices. There is archaeological evidence that this essentially fueled the economy of Jerusalem. The events described in the Bible when Jesus drives the money changers from the Temple are also an example of this - the money changers were there because the faithful had to travel to the temple to make sacrifices, and would need to exchange foreign currency for the local currency in order to purchase animals for sacrifice. By way of comparison, the Christian religion at the time was essentially free. Similarly, the Roman religion could be very expensive - for instance, there is widespread archeological evidence of defixiones, led tablets containing magical spells and curses. Obtaining these, or obtaining defense against such curses, cost money, and so a poor Roman could wind up being cursed and unable to afford any sort of countermeasure. Christianity offered a free sort of defense against such charms. The story of Simon Magus in Acts is an example of this (though Simon was a Samaritan sorcerer and not a Roman tablet maker, the principle is the same) - it emphasized both that the power of the Christian God was greater than that of mortal sorcerers and that money was not required in order to obtain blessings or ward off curses.