Check up is typically used to refer to investigating the progress of something. For example "Let me check up on how my students are doing with their science experiment" Or "Can you check up on my assistant while I'm away" Toaster's explanation of check out is spot on right. I can't think of anything to add to that.

You check up "on" someone. That means you make sure they're okay. A check-up is a routine visit to the doctor's. That's the only way I can think of it really being used. To check out has many more uses. You can check out of a hotel. (Cela signifie quitter l'hotel.) You can check out a book, rollerblades, bowling shoes and many other items. (Cela signifie louer, bien que en la cas du livre, il probablement est gratuit parce que ce serait d'un bibliothèque.) You can check out a song, a movie, and many other things.This is slang. Check out this song = Écoute à cette chanson. And lastly, you can check out a person, more slang. (Cela signifie pour regarder la personne.)