What Anna said lol. "Catch a movie" is meant to sound really casual and not formal at all (you would never write an academic paper or office memo and say it). So you could say, "Hey if you're not busy later, do you want to catch a movie?"
You could use "see a movie" or "watch a movie" in a more formal setting, and while you could ask someone to see a movie casually, it sounds a tiny bit too formal. My mother says "Do you want to see a movie later?", but I don't usually. I would say "watch a movie" but more likely "catch a movie".
Also, this could just be an American Midwest thing but we say "see a movie" in a statement and "catch a movie" in questions.
-"We can see/watch a movie if you're not busy".
-"Do you want to catch a movie later if you're not busy?"