Nothing! They are synonyms! Happen is used much more often than occur, though. Occur sounds more formal, but they mean the same thing :)

As 'nyxielle' said but there are times when they are not the same. Take these sentences: 'The idea never occurred to me' 'It had occurred to me that this wasn't going to be a walk in the park' When occurred is used like this, in the sense of coming to mind, you cannot substitute it with happen :)