They are very close in meaning. Sometimes I hear people use 'clean up' to mean 'tidy up' (i.e., make things orderly), but i never hear anyone use 'tidy up' to mean 'clean up' (i.e., put things in the trash, remove dirt, or disinfect).

I'm Australian, so it very well might be an aussie/british thing.

What about "empty inbox?" Doesn't that term more closely adhere to GTD processes? At first I thought it would be problematic because the inbox doesn't necessarily become completely empty, but then it doesn't become completely clean either.