I mean, I can only shed the light on Christianity as this is the only topic of religion that I am fairly familiar with and I will mostly rely on two works: How our world became Christian by Paul Veyne & Rise of Western Christendom by Peter Brown. Firstly, we need to understand something that was a fundamental difference between ancient roman religions and Christianity, ancient roman religion was manly described by the term "religionis", which mostly means observance of rites & customs. According to Veyne the Romans often had slightly "diplomatic" approach towards religion. They performed the rites and sacrifices and on this basis demanded favours from Gods. Besides that, it was also common that religion, myth and though became quite separated, which created a problem of belief. Christianity on the other hand used term "credo", which means believe, as the very basis of religion. Being a Christian demanded some actual faith in the gospels, one couldn't be a Christian without believing in the gospels. Late Roman pagan on the other was more of a neoplatonist than anything else.

That focus on belief, a story of Jesus & quick use of the ancient philosophy for apologetics created Christianity quite particular and that's why Paul Veyne (who is a disciple of Foucault & Nietzsche btw) described Christianity as the "masterpiece of religious imagination", and we must be aware that the Christianity was a hot topic of debate in the educated Roman circles, also due to works of people like Tertullian. Besides that many thing that were addressed by Christianity were already a common point of antique philosophers and in a way we can say that Christianity is a philosophical sect for the masses. However, we must also be aware that Christianity weren't eschatological destiny (it was also replace by Islam quite easily) and that its victory wasn't really sure and we can surely imagine other turn of events in the case of Julian's victory. However we must also be aware of the advantages that Christianity hold over ancient paganism.

As for early Roman period we must be aware of one thing; the rupture between late roman empire and early medieval Europe was stronger on the economic terms than on cultural terms. Barbarian tribes were already under Roman influence to some extent and Christianity was in a way also a path to the inclusion in the imperial legacy. We can also ascribe similar factors as before which gave Christianity some kind of advantage over the ancient paganism, tho.