In both cases the split occured in part because the existing centralized religion became excessivly corrupt. This sparked a backlash movement that had to find authority and place faith in something other then the central church.
The Islamic side was never as centralized as the Christian side, but they let themselves become a tool of the corrupt and unpopular government of the era. Of course, the Catholic church was a major government in and of itself, with a list of problems that would take a book to list.
Jay