Yes, there's a potential difference.Hello!
Is there a difference between:
Would you like one more?
Would you like another one?
I don't agree. Both can be used whether the person addressed has already had one or more than one."One more" would indicate that you have been given a number of things and are being asked if you would like another unit, whereas "another one" implies that you have a singular item and are being asked if you would like another singular unit.