You can only "find out" information. You can "discover" both things and information.
The difference between "find out" and "discover", when talking about information, has to do with register. "Find out" is a phrasal verb, so it's common, everyday speech. "Discover" has a French/Latin origin, so we see it as a more businesslike, serious and even academic word (even though it is still a very commonly-used word in English).