What is the difference between Islam and Christianity? If Jesus is God, what about the prophet Moses (Torah), or the prophet David (Injil)? The holy book Torah had been given to the prophet Moses and the holy book Injil had been given to the prophet David. Likewise, the holy book had been given to the prophet Jesus and finally the holy book, the Quran had been given to the prophet Mohammed. If you don't accept the Qur'an, how you expect the Jews to accept the Bible? So, it is clear that all the books are the word of God. Whoever lives in the particular period should follow what had been given at that time.
It is clear that you see a difference between Islam and Christianity, just as the rest of the Muslim world does. If there was no difference, then Muslim countries would not declare it illegal for Muslims to convert to Christianity.
The difference is that what David, Solomon, Isaiah, Ezekiel, Daniel, and host of other prophets recorded was all compatible with what Moses wrote earlier. There was no conflict in the teachings nor were there inaccuracies in what was recorded. Jesus brought in a new set of teachings, but these changes were what God said would come through Moses and the other prophets. Jesus' life and teachings matched what was prophesied about him exactly. Like the prophets before him, Jesus backed up his claims with miraculous evidence from God. Unlike the prophets before him, Jesus gave proof that he was God -- but that too was predicted by the ancient prophets.
Mohammed claimed to be a prophet, but no miracles accompanied his claim to show God was with him. The writings attributed to him do not match what was recorded by the ancient prophets in detail or in character. Nor was there predictions that new teachings were to come after the teachings of Christ. Actually, the New Testament clearly states that it is the last covenant between man and God. See "What About Latter-Day Revelation from God?" for details.
By your argument, you ought to become Mormon because Joseph Smith claimed to be a prophet of God and came after Mohammed.