More can always be said on this subject, but I’m going to focus on Emperor Constantine and Roman culture writ large.

To answer why the Romans would be willing to accept another faith, the answer is syncretism. The Roman Empire often incorporated aspects of culture and religion from the places it conquered, thereby weaving together the many disparate populations under its control, a practice first fervently practiced by Alexander the Great. Rome viewed itself as the cultural heir to the Hellenistic culture that preceded it, and several of that culture's gods were favored among Romans (like Isis). Dionysus, a Greek god, was also imported into the Roman canon. There was a practice of incorporating deities into the Roman system, so the idea of adopting another, new god isn't beyond the realm of likelihood.

Another huge factor is Emperor Constantine. Ancient historians Eusebius and Lactantius reported Constantine’s conversion to Christianity. According to Lactantius, the night before the battle, Constantine dreamed that he must “delineate the heavenly sign on the shield of his soldiers.” (Lactantius, On the Deaths of the Persecutors 44.5). He followed the command, and marked his soldiers’ shields with a sign denoting Christ. Lactantius described it as a Latin cross that rounded, but Eusebius described the more well-known Chi-Rho sign. Whichever story, Constantine wins the battle to end the Tetrarchy and become the sole ruler of the Roman Empire. This marks the beginning of Constantine’s conversion to Christianity.

His transformation wasn’t immediate or of whole cloth, however. He didn’t immediately outlaw cult practices for non-Christians or close any temples. On the contrary, he issued the Edict of Milan which legalized all religious practices in Rome. He built churches that he dedicated to the Virgin Mary, but also invited a pagan priest to bless the city. He doesn’t remove the pagan god Sol Invictus from coins until 13 years after Milvian Bridge. According to the pagan writer Zosimus, Constantine later banned private divination and magic because he was afraid others would gain power over him. Constantine had himself made use of soothsayers and pagan prophecies prior to his conversion. The account of Eusebius conflicts with writers like Zosimus, as Eusebius describes Constantine as steadfast and unshakable in his beliefs. He wasn’t baptized, which is a very important tenant of Christianity, until his deathbed. He was the first Christian emperor of Rome, but he was still a member of a largely syncretic society that tolerated a lot of different religious deities and practices. There’s also arguments to be made that at least some of Constantine’s discussions around the Christian God could be mapped onto another syncretic god, Sol Invictus.

I can’t speak to why Christianity over Judaism specifically, but Christianity is unique compared to the followers of other Roman cults in that Christians have proselytization as a central tenant. As Christians, they’re supposed to spread the Gospel. This teeters towards the edges of my understanding of Early Christianity, however, so it’d be best for me not to speak further on it.

Ultimately, Constantine’s religious beliefs are more complex than simply saying “he converted at Milvian Bridge” and, therefore, he’s the reason Romans were willing to accept Christianity. Roman society’s acceptance of shifting and adopting from other culture’s coupled with an at least ostensible Christian emperor, along with Christianity’s proselytization all contribute to Christianity’s growth in the Roman empire.

Citations

Peter Brown, Rise of Christendom 2nd edition (Oxford, Blackwell Publishing, 2003)

Eusebius, The Life of Constantine

Ramsay MacMullen, Christianizing The Roman Empire A.D.100-400 (Yale University Press, 1984)

Ramsay MacMullen, Christianity and Paganism in the Fourth to Eighth Centuries (Yale University Press, 1997)

Constantine, Oration to the Assembly of the Saints