These are two common and useful phrasal verbs that can sometimes be confusing. The general meaning for both of these is quite similar, but there there is one small difference.
Find out means to discover or learn information about something. As a phrasal verb, this is different than the verb find, which means to discover or locate something. We can find out information from various sources: the internet, a book, by asking someone who knows the answer and having them tell us, etc.
- I will find out how late the restaurant is open. (I will call or check the website.)
- Can you find out where the closest bank is? (Google it, check a map, look in a directory, etc.)
- I would like to find out where she bought her car. (I will ask her.)
Figure out means to arrive at an understanding about something simply by thinking about the question or considering the problem, and coming to a logical answer or solution without any outside help.
- I need to figure out what is wrong with my computer. (I will work on it, try several things, and discover the problem on my own.)
- I can’t figure out why she is mad at me. (I have thought about it and I can’t think of the answer, but I haven’t asked her.)
- Can you figure out the answer to this crossword puzzle? (I am asking them to read the clue and see if they can think of the answer for the clue. However, I don’t want them to simply look for the answer somewhere and tell me.)