Example: Girl: Are you serious? Do you really want to break off our engagement? Guy: Yes I want us to break up. I have fallen in love with someone else.

If two people break off a relationship, they break up. You break off a relationship, you don't break up a relationship. You break up with your partner, you don't break off with your partner.

Seems really similar. Almost interchangeable. However. . . . "Breaking up" seems more permanent. "Breaking off" is more of a euphemism to soften the idea, since it is often harsh to admit to oneself that he is actually "breaking up." Speaking of the situation in the past, we'd always say "broken up", and never "broken off". UNLESS, you'd want to say, "We broke it off + time period."

never said )) but i can say it in jest.

to break up = intransitive verb We broke up. (no object) = the relationship ended to break off = transitive verb We broke off our engagement. = we ended our engagement