wrap up vs finish up

wrap up and finish up both are verbs.

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

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

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

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