"Break up" means to separate into pieces or parts. A class can break up for the holidays, or you can break up bread. A couple ending their relationship break up. "Break off" can literally mean that you break the end off something, eg. break off the end of a baguette, or it can mean that you end something suddenly without completing the original plan. For example, you break off an engagement (ie. sudden separation, no wedding) or break off a meeting. If there's an object in the sentence, this also affects the meaning of the phrasal verb.

People break up. You break off a relationship. That's the difference.