What's the difference between "give up" and "give up on"?

Give up is an intransitive verb. This means that there's no gramatical object for the verb. Learning how to play the piano was too hard, so I just gave up. Give up on is a transitive verb. This means that there's a grammatical object for the verb. I gave up on playing the piano because it was too hard. In this case, the object is "playing the piano". In the first case, there is no object.