break off vs hang up

break off and hang up both are verbs.

Word	Noun	Adjective	Verb	Adverb
break off	No	No	Yes	No
hang up	No	No	Yes	No

As verbs, hang up is a hyponym of break off; that is, hang up is a word with a more specific, narrower meaning than break off:

break off (verb)	hang up (verb)
prevent completion	interrupt a telephone conversation
interrupt before its natural or planned end	put a telephone receiver back in its cradle
break a small piece off from	cause to be hanging or suspended
break off (a piece from a whole)
break a piece from a whole