These are both ways of asking the same question, but they are a little different.
The 'Would you like ...?' form is more polite, and is more of an invitation or an offer. It can also refer to any time, such as some time in the future.
'Do you feel like going to a movie?' is talking about now - about whether you're tired or bored or restless or hungry ie what your body is telling you at the moment. So might say, for example:
A: Do you feel like going to a movie? The new James Bond is on at the Roxy.
B: Mmm, I'd like to see it, but I don't feel like going to the movies right now. I'm a bit tired and I'd rather stay home this evening. Let's go another time.
B would like to see the movie, but she doesn't feel like it at the moment. This explains the difference.