Actually, Muslims and Christians DO believe in the same God. It's in the Koran. They believe that Jesus was a prophet whose words were twisted by men and that Mohammad (the Koran's author) was another prophet who spoke directly to Allah (God). Christianity denies that Mohammad was a prophet. That's the main difference between them.

Islam in fact teaches non-violence, but does state that you CAN commit violence to defend your religion (originated around the time of the Crusades actually) . Many dictators and terrorists twist this idea.