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.
Orthodoxy
Orthodox Judaism is not a centralized denomination. Relations between its different subgroups are sometimes strained and the exact limits of Orthodoxy are subject to intense debate. Very roughly, it may be divided between Ultra-Orthodox or "Haredi", which is more conservative and reclusive, and Modern Orthodox Judaism which is relatively open to outer society. Each of those is itself formed of independent streams. They are almost uniformly exclusionist, regarding Orthodoxy as the only authentic form of Judaism and rejecting all competing non-Orthodox interpretations as illegitimate. While adhering to traditional beliefs, the movement is a modern phenomenon. It arose as a result of the breakdown of the autonomous Jewish community since the 18th Century and was much shaped by a conscious struggle against rival alternatives.