Christianity, and by extension the empire, massively changes. the first communities of christians outside judea were diaspora jews, and even for them it took a small schism until they shared the new religion with all the other kids.
Without the diaspora christianity will only have judea to take, and then it has to face judaism - a fellow monotheist, abrahamic religion with none of the flaws which doomed roman paganism, and a huge zealot streak. Best case scenario judaism stomps out the new belief or christianity just becomes an updated judaism like how rabbinic judaism is a continuation of ancient judaism (early christianity was much more jewish and hebrew/aramaic than the catholic/orthodox/protestant faiths today).
Worst case scenario, judea becomes a hotbed for civil war or splits in two altogether - a "notzri" north and a "yehudi" south (side note - interesting idea for a TL).
now for the second part, while the jewish diaspora itself did not massively impact the empire directly (as much as I know), the lack of christianity is HUGE. Maybe sol invictus or any other local/eastern monotheistic faith rises to prominence, but I would like to see how an actually pagan empire proceeds - maybe it becomes like hinduism, with a quasi-ethnic poly-theistic religion dominating all of the mediterrenean basin and "romanising" outside kingdoms.
As of the empire itself, it is quite impossible to tell. Christianity or no, the migration era is still coming. Maybe with the removal of christianity the empire will be less internally discohesive when the huns come, but the potential butterflies border on infinite. Best case scenario for the empire, they settle on the hindu-like religion I mentioned earlier, and manage to achieve stability in the 3rd century earlier through this "religio romana". This means the goths and huns will face a rejuvenated empire with an even better romanisation engine - if you can get that germanic tribe to worship sacred texts in latin and pray to greek, italian and celtic gods, than you've already won.