If you imagine yourself with a piece of paper and some scissors, you would 'cut out' a picture from the middle of a page, but you would 'cut off' a form to sign from the bottom. It's the same with the phrasal verbs, when used in a figurative sense: eg My doctor advised me to cut out dairy foods from my diet. Our electricity was cut off while the building work was being done.

"cut out" would be to remove something from something else, like cutting out a picture from a magazine. "cut off" would be to remove the end of something; "she cut off her hair"