Muslim beliefs are different -- how so if there's one Allah and one Koran?

There's different branches of Islam based on ancient happenings related to who would be the rightful successor of Muhammad. They all believe in the Koran and Allah still, but aren't sure about other things. Kind of like Catholicism, there's different branches of that too, Christian and Catholic being most common. They believe in the same general things but as far as specific beliefs go, they differ.