If by "static class" you mean a class that has only static variables, then they actually can maintain state. My understanding is the that the only difference would be how you access this thing. For example:
MySingleton().getInstance().doSomething();
versus
MySingleton.doSomething();
The internals of MySingleton will obviously be different between them but, thread-safety issues aside, they will both perform the same with regards to the client code.