finish up vs mop up

finish up is not a noun while mop up is a noun.

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

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

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

finish up (noun)	mop up (noun)
	a concluding action

finish up (verb)	mop up (verb)
finish a task completely	finish a task completely
finally be or do something	defeat thoroughly
	to wash or wipe with or as if with a mop