With 'crop up' there is a sense of it happening unexpectedly or without notice. With 'come up' there isn't this sense of unexpectedness or lack of warning.

I would add that "[something has] come up" means that something just happened. It usually is used to mean that you can't do something else because of it: "Hey! I'm sorry but something has come up and I can't come to your party today." Or that it didn't become relevant for a conversation: Q: Why didn't you tell me that Stacy and Mark started dating? A: We were talking about your job so it didn't come up and I forgot to tell you. Sorry!