"Save UP" is a phrasal verb, so you might want to consider it a single unit. To "save up" implies that you're saving, or putting aside, and not spending money deliberately, for a purpose to buy something. If you didn't have a target or a purpose, you could just say that you're saving money. Examples: I don't eat out, I'm trying to save money = I don't want to spend money on restaurants, it's expensive, I eat at home. I'm saving up for a new iPhone = I'm putting money aside deliberately to buy an iphone in the future. I have a purpose. On the other hand, "UP" following a verb conveys the meaning that something must be finished or done completely. I've cleared UP my room (there's no rubbish or dirt in it anymore). I ate UP the pizza (I ate the last piece). I'll settle UP with you next week (I'll return you all the money you've lended me).

Save up means exactly the same thing as to save. Don't worry about the difference because there is no difference.