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.
Haskalah
The spreading of Haskalah affected Judaism as a religion because of how much the different sects desired to be integrated, and in turn, integrate their religious traditions. The effects of the Enlightenment were already present in Jewish religious music and opinion on traditionalism versus modernization. Groups of Reform Jews such as the Society of the Friends of Reform and the Association for the Reform of Judaism were formed because they wanted and actively advocated for a change in Jewish tradition, mainly rituals like circumcision. 
Another non-Orthodox group was the Conservative Jews, who emphasized the importance of traditions but viewed with a historical perspective. The Orthodox Jews were actively against these reformers because they viewed changing Jewish tradition was an insult to God and that fulfillment in life could be found in serving God and keeping his commandments. The effect of Haskalah was that it gave a voice to plurality of views, while the orthodoxy stubbornly preserved the tradition, even to the point of insisting on dividing between sects.