In a Nutshell:
Both faiths have more in common than differences. Similarities include doctrine, beliefs, law that of opposing the doctrine of the trinity, angels and demons and aspects of Judaic law.
Introduction
Islam, Judaism and Christianity arise from the descendants of Abraham, thus have similarity in their teachings. All of them have the one fundamental monotheistic message, of an eternal God, who created the universe and mankind, life is a test to identify those who serve God and those who do not, and a day of judgment that will inquire into how one lived their life and eternal afterlife of bliss or torment.
Similarities between Islam and Judaism
Doctrine and Beliefs:
Islam and Judaism both accept one God with similar attributes as the creator of the universe, without partner or similarity to his creation. Mankind entered into a covenant with God and life is a test of that agreement which will be enquired into on the day of judgement with eternal reward a consequence for accepting and serving God thereby living a good life and hell for a ba life. Both consider the Christian doctrine of the Trinity and the belief of Jesus being God as blasphemous.
Both faiths believe in angels, as servants of God and share a similar idea of demons (Jinn and Shedim). Jewish demonology mentions ha-Satan and Muslim demonology mentions Al-Shai'tan both rejecting him as an opponent of God. Many angels also possess similar names and roles in both Judaism and Islam.
Revelation:
Both share the idea of revealed scriptures and the prophets from Adam, Noah, Abraham, Moses, David and Solomon. Islam confirms Jesus (Isa) and Muhammed as prophets whilst Judaism rejects them.
Muslims, however, believe Jewish scriptures were distorted and corrupted over time, with significant textual alteration and contextomy, so were abrogated by revelation to the final prophet of God Muhammed (saw), namely the Qur'an and Sunnah.
Law:
Both faiths subscribe to revealed law, seeing no separation of divine and profane. In Islam, the laws are encompassed by the Shari'a that addresses morality, ethics and law, and in Judaism, the equivalent is the Halakha. Both consider the study and practice of law to be worship or servitude of God.
Both have in common notions of prayers, fasting, almsgiving, ritual purity and dietary laws which includes the prohibition of pork. Jews, for instance, pray in Hebrew, Brachot in the morning until the first quarter of the day, Shacharit from sunrise to noon, Ashrei/Mincha in the afternoon, Maariv at night and Shema before sleeping. Muslims pray Arabic, Fajr at dawn, Zuhr afternoon, Asr midafternoon, Mahrib at sunset and Ishaa at night.
Neither faith subscribes to the concept of original sin and both religions view sexual relations outside of marriage including homosexuality and adultery as sinful. Society is structured as patriarchies in both faiths, polygyny is permitted, however within contemporary societies it is not a widespread practice. Dress codes exist for females, and modesty is expected from both males and females.
The Islamic Hadith and Jewish Talmud have also often been compared as authoritative extra-canonical texts that were originally oral transmissions for generations before being committed to writing.
Differences between Islam and Judaism
Despite similarities, that are significant differences. Jews do not accept the final prophet Muhammed, his revelation or the wider social and political arrangements he brought.
Conclusion
All Abrahamic traditions have significant similarities, with the same underlying message of one God who we are all expected to serve, part of which has been forgotten or distorted over time.
