Because Christians are not a homogenous group: trying to pin down the total list of beliefs for all of the billions of Christians around the globe is rather hard, not just now, but throughout history. Christina's have been divided on any number of beliefs or opinions and the history of the religion is rife with divisions and enmity that carry forward for centuries.

While Rome "persecuted" the early church, the Empire under Constantine was ultimately responsible for codifying it and expanding it throughout the Empire. This is reflected by the highest position of the Roman Catholic church being the Bishop of Rome who is also the Pope, but the Roman Catholic Church is also a result of the split between the Bishop of Rome and the Bishops of the Eastern Orthodox churches.

The persecution of Christianity is somewhat over dramatized. Rome by today's standards was fairly brutal to dissidents. The early Christian church was often conflated with Jewish rebellion which resulted in the destruction of the second Temple in 70 CE and the diaspora of the Jews. The persecution of Christians was no different to the Romans than it was for the Jews, or to any other different group they had under their empire who seemed to challenge the Roman way of doing things. As Christianity rose to popularity it was seen as a challenge to traditional Roman beliefs, but Emperor Constantine used the popularity of it to his favor and subsumed the cult. Christianity in many ways became a Roman religion under Constantine instead of a Jewish cult.

Further compounding this is the religious belief of some modern Christian groups who see the religious act of crucifixion of Christ as being required as a matter of the religion to the point that the crucifixion serves as a form of religious sacrifice for all people for all time, so that the sin of an individual required the sacrifice of Christ to do it.

However throughout the centuries the desire to punish Jews for the death of Christ would have been more common as an excuse for anti-semitism. Within the story of the crucifixion the Jewish leaders are responsible for having Jesus arrested and charged, and the Roman Leader offered to free a prisoner either Jesus or another person and the Jews didn't select Christ.

It's also important to note since this is ask historians there isn't historical record of the Crucifixion of Jesus of Nazareth, who was also entombed in a tomb meant for someone else. Only a single Roman historian notes Jesus but the reference was more about the rising cult, and doesn't independently confirm the historical person of Jesus. Which makes sense. Jesus in the context of contemporary historians would have been a relatively minor cult leader who preached for only a few years before being killed. He lead no armed rebellions, caused no major battles, and would barely appear on the radar under the Roman cultural values of greatness. There would have been any number of similar cult leaders across the Empire. And I say this as a Christian who recognizes the value of the historical record. And objective view of the Judeo Christian religious texts in context with the current historical record results in few independent confirmations across history, which is a much a reminder of how little impact most lives make on history and how little ultimately survives to be remembered properly over thousands of years.

I'm sorry this is so late, but I actually specialize in Late Antiquity and I get this question a lot. The other answer is quite good (except for the assertion that we have no historical references confirming the crucifixion of Christ-we Classicists have argued a million times about the interpolations and emendations in Jospehus, but we almost all agree at the this point that his biographical references to Christ and John the Baptist are at least partially authentic). But I have to add that much of Rome's importance in Christianity is associated with the idea of apostalic succession, martyrdom, and the cult of saints.

Even before Christianity was legal, Christians looked to the bishop of Rome as something like an authority among the bishops. Theologians and Classicists argue about just how much authority the pope had, but he certainly had some authority over communities which were outside of his own purview. The First Epistle of Clement 1 is a very famous and controversial text which is frequently cited as evidence of the pope's global authority. While the date of composition is uncertain, it was certainly written before around 150 AD, well before the edict of Milan in 313 and thus safely within the persecution period.

So why did the pope have this authority? Well, the simple and common answer is that it is the only bishopric founded by St. Peter, who received the authority to found the Church directly from Christ. But, over and on top of that: Both Peter and Paul were martyred and buried in Rome. As these were two of the most revered apostles, the fact that they both participated in building up the church in Rome was itself very important. As Saint Ireneaus said probably around 200 AD: "...we shall turn to that great, ancient and universally known church founded and organized at Rome by the two most glorious apostles Peter and Paul, and we shall show that the tradition it has received of the apostles and the faith that it preaches to men has come down to our time through the regular succession of its bishops..."

Back to the martyrdom thing. It is basically impossible to give a succinct summary of the theology and social impact of martyrdom in Late Antiquity, but all you need to know is that martyrs were revered among early Christians and that their power was considered to be highly local, i.e. their power was greater the closer you were to their remains or the place they suffered and died. In practice, this meant many early Christians liked to hold worship services in catacombs or graveyards so they could be close to the martyrs' bones. The dust which absorbed a martyr's blood could even be collected and preserved (there is a great example from Prudentius's Crown of Martyrs in the Fructuosis poem, but I can't find a free version). This is the same logic, by the way, which is behind the modern Catholic reverence for relics. So, because Peter and Paul were both martyred and buried in Rome, and they were the "most glorious apostles," Rome became a very holy place which Christians would have considered a very fitting place to hold *extra holy* services.

Edited to add a nice summary (because I was rushed when I first wrote this): So then, Rome was considered important among the pre-edict of Milan Christians because the church there was founded by Pete himself, which is the closest one can get to Christ himself founding a church (Mt. 16:19); because St. Paul, the other great Biblical apostle, worked closely with the church there; and because both Peter and Paul died and were buried there. I think the other answer provides a lot of good info about Rome's importance post-persecution period. The only thing I'd add is that while mystery cults were a huge thing at this time throughout the empire, there was something either socially or theologically attractive about Christianity which gave it a foothold among middle- or upper-class women, which is why it ended up becoming legal (via Helena, mother of Constantine). There are so many great articles and chapters I want to share with you on this issue, but sadly none of them are free. Suffice to say, though, great question! Thank you so much!