They are equivalent to each other. However, to my ears (as a native speaker of American English), finish up sounds better in this context.
More examples:
John finished up the dishes after dinner.
John finished off a bottle of wine before going to bed.
Both have the sense of completion, but there is a subtle difference in feeling between the two. Finish off feels much stronger, which is perhaps why it can also be used to mean to kill something.
It was the blow to the head that finished him off, not the bullet wound.
Incidentally, Finish up is prefered intransitively, whereas Finish off is much rarer, at least in American English.
This sounds more natural than using finish off:
Finish up so we can go.
The race finished up in Boston.