The existence of potatoes in the ancient European setting of Middle-earth is problematic since potatoes came from America only in the recent centuries. Tolkien avoided using foreign concepts and terms of English vocabulary such as Native American words. For example tomatoes were referred in the first edition of The Hobbit but removed in the second one.
The term "taters" is perhaps an attempt to 'anglicize' or 'Hobbitize' this word.
Conceivably, the roots named 'potatoes' in Tolkien's work represent some other related root plant, or a crop brought east by the Númenóreans that later became extinct - similar to Tolkien's explanation of how Pipe-weed was introduced to Middle-earth.