polish off vs finish up

polish off and finish up both are verbs.

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

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

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

polish off (verb)	finish up (verb)
finish a task completely	finish a task completely
finish eating all the food on one's plate or on the table	finally be or do something
kill intentionally and with premeditation