Favorite and favourite are different spellings of the same word. Favorite is the preferred spelling in the U.S., while favourite is preferred in all the other main varieties of English. These preferences extend to all derivatives, including favorites/favourites, favoritism/favouritism, and favorited/favourited.
Favourite has been the preferred spelling in British English for several centuries, but this does not mean that favorite is a late arrival to the language or even American in origin. In fact, the OED lists instances of favorite from as long ago as the 17th century. Milton used it in Paradise Lost and, and William Wordsworth used it around 1800 (and many more examples are easily found in historical Google Books searches). The preference for favourite was not well established in British English until the first half of the 19th century, which, as the below ngram shows, is around the same time Americans settled on what is now their spelling.
Point, set, match.