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"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.
eg: "Would you like one more biscuit? You have already eaten ten."
"You seem to enjoy the slice of cake. Would you like another one (another slice)?"
Yes, there's a potential difference. Customer: This pineapple is bruised.
Shopkeeper: Would you like another one? The shopkeeper isn't asking if the customer wants two pineapples. He's asking whether the customer wants an alternative non-bruised 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.