Anything is basically saying 'any one thing'. Just one. I'm sure, however, that there are probably instances where it might be more than one thing.

Everything should be take word for word, 'every thing'.
(Apologies for my suckish examples...)
If you were to walk into a store and say, "I want everything you have." That would mean every single thing in that store, versus 'anything' in that same sentence. Or maybe if it is in that sentence, it almost sounds like it means more than one thing, but it shouldn't.

I think (: