I have some friends who are Messianic Jews, but most of them started out as Christians, not as Jews. When I was in college I knew some “Jews for Jesus” who started out as Jews and then decided Jesus was the Messiah. Their services are quite different from each other, and also different from typical Jewish services.
As to why Jesus is not the Messiah, per this site:
1) JESUS DID NOT FULFILL THE MESSIANIC PROPHECIES
What is the Messiah supposed to accomplish? The Bible says that he will:
A. Build the Third Temple (Ezekiel 37:26–28).
B. Gather all Jews back to the Land of Israel (Isaiah 43:5–6).
C. Usher in an era of world peace, and end all hatred, oppression, suffering and disease. As it says: “Nation shall not lift up sword against nation, neither shall man learn war anymore.” (Isaiah 2:4)
D. Spread universal knowledge of the God of Israel, which will unite humanity as one. As it says: “God will be King over all the world—on that day, God will be One and His Name will be One” (Zechariah 14:9).
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2) JESUS DID NOT EMBODY THE PERSONAL QUALIFICATIONS OF MESSIAH
A. MESSIAH AS PROPHET
Jesus was not a prophet. Prophecy can only exist in Israel when the land is inhabited by a majority of world Jewry. During the time of Ezra (circa 300 BCE), when the majority of Jews refused to move from Babylon to Israel, prophecy ended upon the death of the last prophets—Haggai, Zechariah and Malachi.
Jesus appeared on the scene approximately 350 years after prophecy had ended.
B. DESCENDENT OF DAVID
The Messiah must be descended on his father’s side from King David (see Genesis 49:10 and Isaiah 11:1). According to the Christian claim that Jesus was the product of a virgin birth, he had no father—and thus could not have possibly fulfilled the messianic requirement of being descended on his father’s side from King David!
C. TORAH OBSERVANCE
The Messiah will lead the Jewish people to full Torah observance. The Torah states that all mitzvot remain binding forever, and anyone coming to change the Torah is immediately identified as a false prophet. (Deut. 13:1–4)
Throughout the New Testament, Jesus contradicts the Torah and states that its commandments are no longer applicable. (see John 1:45 and 9:16, Acts 3:22 and 7:37)
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3) MISTRANSLATED VERSES “REFERRING” TO JESUS
Biblical verses can only be understood by studying the original Hebrew text—which reveals many discrepancies in the Christian translation.
A. VIRGIN BIRTH
The Christian idea of a virgin birth is derived from the verse in Isaiah 7:14 describing an “alma” as giving birth. The word “alma” has always meant a young woman, but Christian theologians came centuries later and translated it as “virgin.” This accords Jesus’ birth with the first century pagan idea of mortals being impregnated by gods.
B. CRUCIFIXION
The verse in Psalms 22:17 reads: “Like a lion, they are at my hands and feet.” The Hebrew word ki-ari (like a lion) is grammatically similar to the word “gouged.” Thus Christianity reads the verse as a reference to crucifixion: “They pierced my hands and feet.”
C. SUFFERING SERVANT
Christianity claims that Isaiah chapter 53 refers to Jesus, as the “suffering servant.”
In actuality, Isaiah 53 directly follows the theme of chapter 52, describing the exile and redemption of the Jewish people. The prophecies are written in the singular form because the Jews (“Israel”) are regarded as one unit. The Torah is filled with examples of the Jewish nation referred to with a singular pronoun.
Ironically, Isaiah’s prophecies of persecution refer in part to the 11th century when Jews were tortured and killed by Crusaders who acted in the name of Jesus.
From where did these mistranslations stem? St. Gregory, 4th century Bishop of Nanianzus, wrote: “A little jargon is all that is necessary to impose on the people. The less they comprehend, the more they admire.”
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And finally: Of the 15,000 religions in human history, only Judaism bases its belief on national revelation—i.e. God speaking to the entire nation. If God is going to start a religion, it makes sense He’ll tell everyone, not just one person.