finish up vs wrap up

finish up and wrap up both are verbs.

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

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

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

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