Jesus Christ, the undisputed founder of Christianity was undoubtedly Jewish, both ethnically and religiously. He never set out to start a new religion but to fulfill the old. [Matthew 5:17] So why are today’s Christians not considered Jews?
Clearly there are some major differences between Judaism and Christianity:
Christianity is rooted in Second Temple Judaism, but the two religions diverged in the first centuries of the Christian Era. Christianity places emphasis on correct belief (or orthodoxy), focusing on the New Covenant as mediated through Jesus Christ, as recorded in the New Testament. Judaism places emphasis on right conduct (or orthopraxy), focusing on the Mosaic Covenant, as recorded in the Torah and Talmud.
Christians believe in individual salvation from sin through repentance and receiving Jesus Christ as their God and Savior through faith (and in some Christian traditions, good works). Jews believe in individual and collective participation in an eternal dialogue with God through tradition, rituals, prayers and ethical actions. Christianity generally believes in a Triune God, one person of whom became human. Judaism emphasizes the Oneness of God and rejects the Christian concept of God in human form. [Wikipedia]
However, one could argue that perhaps just as much difference exists between Mormonism and Catholicism yet both denominations still identify as Christian.
I think it is primarily an issue of self-identification. Today, and starting as early as the first century AD, Christians did not necessarily want to identify with the people they believed “killed Jesus”. For early Gentile converts, converting to Judaism also meant painful rituals and intensive training in Old Testament law. If a Gentile could easily become a Christian and bypass all of that, why bother?
The Gentile Christians soon began to outnumber the ethnically Jewish Christians and so Christianity was no longer seen as a primarily Jewish religion. Judaism has always been seen as an essentially ethnic religion, after all.
Presumably, today, you can still find people who still identify as both Jewish and Christian, however, the ‘Jewishness’ to which they refer is probably primarily that of ethnicity.