Judaism
Christianity was originally a sect of Second Temple Judaism, but the two religions diverged in the first century. The differences between Christianity and Judaism originally centered on whether Jesus was the Jewish Messiah but eventually became irreconcilable. Major differences between the two faiths include the nature of the Messiah, of atonement and sin, the status of God's commandments to Israel, and perhaps most significantly of the nature of God himself. Due to these differences, Judaism traditionally regards Christianity as Shituf or worship of the God of Israel which is not monotheistic. Christianity has traditionally regarded Judaism as obsolete with the invention of Christianity and Jews as a people replaced by the Church, though a Christian belief in dual-covenant theology emerged as a phenomenon following Christian reflection on how their theology influenced the Nazi Holocaust.
Jews
Without a Temple, Greek-speaking Jews no longer looked to Jerusalem in the way they had before. Judaism separated into a linguistically Greek and a Hebrew / Aramaic sphere. The theology and religious texts of each community were distinctively different. Hellenized Judaism never developed yeshivas to study the Oral Law. Rabbinic Judaism (centered in the Land of Israel and Babylon) almost entirely ignores the Hellenized Diaspora in its writings. Hellenized Judaism eventually disappeared as its practitioners assimilated into Greco-Roman culture, leaving a strong Rabbinic eastern Diaspora with large centers of learning in Babylon.