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."

They can both be adjectives and verbs. Here's how they're used as adjectives: As adjectives, they almost the same. The only difference that I can think of as that saying something is tidy implies that someone tidied it, while something that is clean could be like that naturally. For example, I could say that grass looks clean or that water from the tap tastes clean, but I'd never say they're tidy.

I seldom use the word tidy. When I hear that word my mind sees a room with orderly stacks of things that should be discarded. Neatly stacked piles can be tidy but dusty or even dirty. I prefer a clean room with disorder over tidy.