The first Roman attacks on Jews came in the 1st century BCE, before Christ and before the first rebellion against Rome (66 AD). Apion and Cicero being among the earliest. The original Roman view of Christianity would have been as yet another squabbling Jewish sect. But slowly they were seen differently. Even before the first Jewish revolt Emperor Nero singled out Christians, not Jews. I am not sure why. At the time Jews proselytized, but perhaps Christians were more irritating about it so got treated differently.

After the second Jewish Rebellion (against Hadrian) Roman attitudes became even harsher towards Jews... but Jews also could be completely tolerated. Still, the center of Jewish culture shifted away from Roman territory to, ironically, the Babylonian region of the Parthian (later Persian) Empire. When Rome became Christian, the treatment of Jews declined even more in a tradition that lasted until the reformation. Often, protestant groups were more tolerant than Catholic groups. Bottom line, though, anytime church and state get blurred too much, Jews don't do so well.