What are the main differences between the Rinzai and Soto Zen schools? Is it right that koans are only used in one of the schools? Also are there other Zen schools and if so what distinguishes those schools from the other Zen schools?
The most important differences between the two schools have to do with their style of meditation and ideas surrounding the attainment of enlightenment. Soto takes a more gradual approach - kind of like a flower opening. Just like a flower coming into bloom, it makes no sense to force anything. Under the right conditions, and when cultivating tenderly, the mind will open of its own accord.
Conversely, Rinzai puts more emphasis on the possibility of sudden enlightenment. One of the easiest ways to consider how koans work in this process is to think of samsara like a brick wall. Investigating koans is like knocking a brick out. A certain amount of light shines through once you see into an individuals koan's 'meaning'. Eventually, like a brick wall, once you knock enough bricks out, the whole thing just suddenly collapses. It's kinda like going broke - it happens gradually and then suddenly! ;-)
Of course, a lot of Zen schools do incorporate elements of both practices. Sometimes, a school with a more Soto emphasis may assign a student a koan. It might also go the other way too, but that hasn't been my experience.