The meaning is more or less the same, so you can use all of them interchangeably without making a mistake. The difference is mostly in style. It's difficult to describe, but as a first attempt I'd say "warum" is the most "literary" and asks for a general reason, "wieso" asks for a more specific reason, and "weshalb" asks for some sort of agent as a reason. But that's just my vague gut feeling at the moment and may not be entirely correct. Maybe one can pin down the differences by looking up the etymology and enough examples.