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Thomas Ryan, CSP Answers:
Here are several ways in which the two religions differ:
Incarnation: The big difference is what Christian theology calls “the Incarnation”, or the “enfleshment” of God as one of us in the person of Jesus of Nazareth.
Trinity: Closely related to this is the revelation of God as a community of relations—Father, Son, and Holy Spirit. If one looks at Jewish mysticism the idea of God being internally dialogical is not so strange. Jewish mystical tradition hold a vision of God as ten-fold, the sephirot emanating from the eternal One, the wholly other. By the last of the emanations, God is approachable by humanity.
Scripture: We share the bulk of Sacred Scripture, but not the New Testament which Jews would see as, at best, something like a rabbinic commentary (midrash) on their Bible.
Messiah and End-Time: Jews do not accept Jesus as their Messiah, but Catholics and Jews share the same vision of the destined End, i.e. of the meaning of human existence. Whether this will involve the coming or return of the Messiah, we both believe it will happen. We must work together to prepare the Way for the Coming of the Messiah and all that it signifies: the end of war, poverty, famine. There will be a bodily resurrection of the dead and a final judgment of all humans. God will be at once fully just and fully merciful. We can and must witness to it together, which means not just words but deeds. We do this by working for justice and peace.
History: Catholics have held power over Jews since the age of Constantine, and have often, though by no means always, abused it. We need to work with Jews to develop a common understanding of our shared, sometimes joyful, too often tragic, common past in which we have learned from each other despite our disputations.