In this context, they have basically the same meaning. If you want to be precise about the meanings of the words though, there is a slight difference.
"Tidy" means to make the room more orderly. If you have objects scattered around the room, you can put them somewhere so that they aren't in someone's way.
"Clean" means to get rid of dirt, such as by dusting, vacuuming or washing windows.
As I said though, most people don't make the distinction when they use these phrases. Either one can be generally understood to mean "make your room look nice."