In a Nutshell:
Both beliefs are based on monotheism however but later credal distortions occurring in the Babylonian Era have changed many fundamentals of Judaism and created significant differences with Islam.
Judaism is the first Abrahamic religion, the founding prophet being Moses, chosen by God to lead the Israelite slaves out of Egypt.
Camped under Mount Sinai, Moses gave the Israelite slaves the Torah, and after wandering in the desert for 40 years, they went to live in what is now known as Israel, which they believe is God's gift to them. Jews trace their Abrahamic lineage through his son, Isaac.
Islam, on the other hand, is the last religion in Abrahamic tradition and was the continuation of God's covenant with the new Prophet Muhammad in the year 622 after the Prophet Jesus. Muhammad wrote down the revelations, he received from God and it was named as Qur'an. Muslims trace their Abrahamic lineage through his son Ismael.
Both are the Monotheistic religions from the core, beliefs in the divine decree and in the existence of angels as a creation of God and Jerusalem as an important as well as common Holy Site. To read more on their similarities, refer to this link.
Key Differences between Islam and Judaism
In Islam, the belief in an eternal afterlife, heaven and hell, is a compulsory credal aspect. Judaism however does not have an agreed upon stance; various views argue no afterlife at all, Olam Haba (world to come), Gehenna (a place of spiritual purification for souls) and reincarnation.
In Islam, God is not like his creation, neither begets nor is begotten. The Yemenite Jewish tradition however claimed Uzayr as the son of God. They later died out and almost no current strands of Jews either orthodox or other denomination believes that.
Belief in Prophets
Islam confirms Jesus is a prophet sent by God, born to the virgin Mary (Mariam). Jesus wasn't crucified nor did he die but ascended bodily to Heaven. Judaism rejects him as a prophet accepting the crucifixion of the historical Jesus. Their Messiah has yet to come and is expected in end times.
Muslims accept the Prophet Muhammad (saw) as the last and final seal of the prophets, whereas Judaism rejects him.
Furthermore, Muslims see all Prophets as sinless, whilst Judaism claims many Prophets are guilty of heinous crimes and sins.
Islam confirms the Torah and Gospel as originally revealed scriptures but have been corrupted. Judaism however confirms The Tanakh/Hebrew Bible (The Torah = Old Testament) but rejects the New Testament and the Qur'an.
Both traditions have a significant number of similarities with some key differences.
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