It's psychological. As soon as you put something on FB, Twitter, etc. it's no longer "yours". Snapchat makes conversations and pictures disposable. It's just fast and ugly. And it powers the app by nudging by users into creating content to send to a friend without having to worry about rehearsing for the perfect picture/tweet/etc.

Also it's private. Facebook is mainly used by people with friends and family. Celebrities/notable figures rarely interact with others outside of Twitter. But even with Twitter, the interactions are permanent -- even when tweets are deleted.

On and it's great for sexting, which let's not kid ourselves, was the original appeal and still is for a lot of people in their target demographic.