wrap up vs finish off

wrap up and finish off both are verbs.

Word	Noun	Adjective	Verb	Adverb
wrap up	No	No	Yes	No
finish off	No	No	Yes	No

As verbs, wrap up and finish off are synonyms defined as:

Other synonyms of wrap up include clear up, finish up, get through, mop up, polish off.

wrap up (verb)	finish off (verb)
clothe, as if for protection from the elements	finish a task completely
form a cylinder by rolling
finish a task completely
arrange or fold as a cover or protection