I personally don't think Christianity comes out on top without imperial sponsorship. But the Plague of Cyprian and the crisis of the Third Century had destabilized the religious situation. According to Historian Kyle Harper, Christianity was invisible from an archeological standpoint until the 250-60s, and traditional religion seemed solid until about that time.
Doesn't mean they don't exist, but think about how society was more default Christian in 1970 vs 2023. Also, think how in 1970 if you asked someone what a Wiccan was in 1970 where you'd probably get a blank look vs 2023 where most people have at least heard of them, you get some idea of how the profile of Christianity rose between 250 and 300. This isn't exact and I wasn't there, but you get the point.
But let's say the empire divides more permanently after Diocletian and a Christ-friendly candidate ends up on top in the East (could happen) and pushes it as an Imperial cult and eventually, say over a century, or so becomes dominant/exclusive. I think the divide between East and West becomes more stark sooner and among the elites there is probably some immigration. In addition, Western Religion continues to evolve.
That division probably strengthens the West as Eastern Emperors still have problems like the Sassinids and the Goths. Also the East, with its denser population get hit by plagues harder and falls further. The Eastern Emperors are busy and the West is more and more alien and the least threat, so things like the battle of Frigidus don't happen, which devastates Western manpower and they do better than otl.
Year 500, The West has lost Britain and most of Gaul and the East has lost large parts of the Balkans, but both are holding a central core. Most importantly Africa is Western. But the East and Western Empire increasingly see each other as foreign and while willing to maintain the fiction of being one Empire to not have to fight on another front. In the West, the religious establishment has figured out this Christianity thing and there is more institutional resistance to the spread of Christianity to the West. After that, the Butterflies become too thick to see.