To "cut someone off" means to interrupt them while they're speaking. "Interrupting" someone is more general. You can interrupt someone watching a TV show. But "cut ___ off" is used specifically to talk about interrupting someone's speaking. You can "cut off" just a single sentence or an entire story or conversation:
Why do you always cut me off like that when I'm talking?
Sorry, I'm going to have to cut you off there. I've got to run.