From what I understand about the topic, Christianity began as an offshoot of the Jewish Abrahamic religion that held the rabbi Jesus to be a messiah sent from god. This by itself is not unique, however, because many other jews from the time period and since have been so acclaimed, ex: Messiahs.

My question is: when (and for what reasons) did it become to be clear to an ordinary Roman pagan for example that a group of ordinary Jews and a group of Christians were distinct and unrelated? Did this become evident once the church had formed its own self-appointed power structure of priests, bishops, patriarchates, etc.? For any church scholars, what were the major turning points in christian theology in particular that separated it from mainstream Jewish theology and who made those contributions?