Smokey's correct name is Smokey Bear. In 1952, the songwriters Steve Nelson and Jack Rollins had a successful song named "Smokey the Bear". The pair said that "the" was added to Smokey's name to keep the song's rhythm. This small change has caused some confusion among the public ever since.
Note that, from the beginning, Smokey's name was intentionally spelled differently from the adjective smoky. The Forest Service emphatically denies that the name was ever "Smokey the Bear"; however, during the 1950s, that variant of the name became widespread both in popular speech and in print, including at least one standard encyclopedia.
A 1955 book in the Little Golden Books series was called Smokey the Bear and Smokey calls himself by this name in the book. The campaign to remind the public of the correct version of the name is almost as old as the Smokey Bear campaign itself.