What are the historical reasons for the different beliefs between Jews and Christians?

Answer 1

The difference began during the lifetime of Jesus, when he preached and practiced in a way not in keeping with traditional Judaism.

However, Judaism speaks very little about Jesus. According to our tradition, the vast majority of the Jews at the time didn't hear of him. The Torah-sages (Rabbi Yochanan ben Zakkai, Rabbi Yonatan ben Uziel, Chanina ben Dosa, Bava ben Buta, Shimon ben Hillel, Rabbi Eliezer, Rabbi Yehoshua, Rabbi Akiva, and hundreds of others) were active at that time and their yeshivot (Torah-academies) were flourishing. Their tens of thousands of disciples and hundeds of thousands of sympathizers were active in the Jewish world in that generation; they were the leaders and the forefront of Judaism. As Josephus (Antiquities book 18) writes, "the cities give great attestations to them." The great majority of Jews loved their sages and their Torah.
The unlearned class of the Amei-haaretz (ignoramuses) was a small fringe of society, but even they would and did lay down their lives in order not to violate anything of the Torah. As one ancient historian famously wrote:
Hecateus declares again, "what regard we (Jews) have for our laws; and we resolve to endure anything rather than transgress them." And he adds: "They [Jews] may be stripped on this account, and have torments inflicted upon them, and be brought to the most terrible kinds of death, but they meet these tortures after an extraordinary manner, beyond all other people, and will not renounce the religion of their forefathers."
No one (even any who did hear of Jesus) - would have given heed to what was and is considered unacceptable for us. The few who came in contact with him soon lost interest, and the early Christians felt the need to turn to non-Jewish centers of population in order to gain adherents, while the Jews remained Jews.
We do not believe that Jesus is or was anything other than a regular human being. We may also note that according to our tradition, prophecy ceased about 340 years before the birth of Jesus; and public miracles stopped even earlier.
Here is a related topic:
The word "messiah" is the transliterated form of the Hebrew "moshiach." The word moshiach translates to "anointed." The title of moshiach was given to any person who was appropriately anointed with oil as part of their initiation to their service of God. We have had many meshichim (plural) in the form of kings and priests. There is absolutely nothing supernatural about a moshiach.
This being said, there is a prophecy of a future moshiach; however, this is a relatively minor topic in Judaism and the Tanach.
The Jewish requirements of the messiah are:
* Build the Third Temple (Ezekiel 37:26-28)
* Gather all Jews back to the Land of Israel (Isaiah 43:5-6)
* 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 any more." (Isaiah 2:4)
* 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)
* The messiah must be descended on his father's side from King David (Genesis 49:10 and Isaiah 11:1)
* The messiah will lead the Jewish people to full Torah-observance. The Torah states that all of its mitzvot (commands) remain binding forever.

Another View:

For some in Christianity, there is a scriptural reason for these differences noted above as having begun during the ministry of Christ. The Apostle Paul told us of it in Romans chapter 9, 10, and 11. He summed it up again in 2 Corinthians 3:12-18. Futhermore, some Christians will also point to Jesus' telling of the Parable of 'The Rich Man and Lazarus' in Luke 16:12-31 where they believe the Rich Man represents the Nation of Israel (primarily Judah) and the beggar represents the 'new branch' called the Gentiles. Abraham tells the Rich Man (Jewish people) that his brothers would have to recognize the prophesied Messiah because of the things written about Him in the Tanakh. This was echoed by Jesus telling the Jews this:

John 5:45-47

New King James Version (NKJV)

45 Do not think that I shall accuse you to the Father; there is one who accuses you-Moses, in whom you trust. 46 For if you believed Moses, you would believe Me; for he wrote about Me. 47 But if you do not believe his writings, how will you believe My words?"

For these Christians then who believe the Jewish people had simply failed to make the association between the words and actions of Jesus to their own teachings. The Scriptures show that Jewish leaders of Christ's day generally failed to recognize the very one Moses wrote about (Deu. 18:15, 18):

Deuteronomy 18:15

New King James Version (NKJV)

A New Prophet Like Moses

15 "The Lord your God will raise up for you a Prophet like me from your midst, from your brethren. Him you shall hear,

As Christ was born a Jew and the House of Judah was given the first opportunity to receive the Messiah, some in Christianity associate Paul's words from Romans 11:8 ("God has given them a spirit of stupor, Eyes that they should not see And ears that they should not hear, To this very day.") as the fate that was to fall upon the Jewish nation because of their unbelief and faithlessness which has led them to reject Jesus as the Messiah. But at the time of the 2nd Coming, God will show His great mercy to them:

Zechariah 12:10

New King James Version (NKJV)

Mourning for the Pierced One

10 "And I will pour on the house of David and on the inhabitants of Jerusalem the Spirit of grace and supplication; then they will look on Me whom they pierced. Yes, they will mourn for Him as one mourns for his only son, and grieve for Him as one grieves for a firstborn.