mop up vs finish up

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

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

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

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

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

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