The authors of Islam took bits of Judaism, Christianity, Zoroastrianism, and quite a bit of the cultural aspects from the inhabitants of the Arab peninsula to create their synthetic religion so that they could first control the locals, and then conquer their neighbors.
One could say that Christianity went through something similar with the various councils that declared some beliefs heretical, etc. But in the long run, Christians have brought about a new enlightenment of our faith by seeking more personal relationships with God....
Plus, when it comes down to it, Jesus’ message is one of peace and personal and individual choice. Islam’s message is one of collective conquest and submission, and it is absolutely that way to its core.
(Islam is flawed at its core, and will be a thorn in the side of humanity so long as a copy of the Koran still exists. But being who we are, we don’t have the will or desire to wipe out an entire religion and its believers just because they want to do the same to us... Our only option is to remain so strong that Islam can never achieve their goals.)
The muslims don't deny at all that the two are similar. They just claim that the bible was repeatedly misinterpreted over the years and that the koran was the correct version and came directly from god.