In Bible, Matthew, what are differences between the jewish expectations and Jesus' claim about the kingdom of heaven and the Messiah.

The book of Matthew starts with the geneology of Jesus
because this was very important to the Jews. The Jewish OT prediced that the Messiah
would come from the line of David, so it was very important to show how Jesus fulfilled
this prophecy. This is the first of the 4 gospels and focuses heavily on how Jesus
fulfilled the Jewish prophecies.

The Jews did not believe
it was proper to mention the name of God, so in this gospel, Matthew observes the Jewish
tradition of using the phrase "Kingdom of Heaven" instead of "Kingdom of
God."

Matthew’s goal was to show that Jesus was the
promised Messiah, the one that Moses spoke about in the law and the one of whom the
Jewish prophets wrote about. There are many allusions to the Old Testament in this book
and many, many more direct quotations from the Jewish Old Testament (the books of Moses)
than any other book in the New Testament. Matthew tries to show that Jesus came NOT to
destroy the Jewish law, but to fulfill it. Jesus is presented as the Messiah, the heir
to the throne of King David, the rightful King of the Jews. The genealogy, the story of
the wise men, the killing of the innocents and the flight into Egypt were all meant
establish Jesus as the predicted Messiah of Israel.

The
Jews did not and do not accept Jesus as Messiah. They beleive Messiah is still to
come. Jesus is the “stone that the builders rejected.” The Jews at the time were hoping
Messiah would come and free them from Roman rule. They expected a savior on a white
horse, which is also predicted in the Old Testament. Christians believe that Jesus will
come on a white horse also, but it will be the second time he comes. Jesus did not meet
the expectations of the Jews when he came as a humble servant and was born in a
stable.