The case
Only you can prevent forest fires, and only a jingle can rename a mascot. The 1952 song by Steve Nelson and Jack Rollins needed an extra syllable, so "Smokey the Bear" went out over the radio, and the ad campaign never fully got it back.
The posters, the PSAs and the Forest Service have always said Smokey Bear. The "the" is a ghost left by a song.
Why our brains do this
A catchy song is a stronger memory than a poster, and the song changed the name for rhythm.
Sources: Wikipedia — Smokey Bear · Wikipedia — False memory: the Mandela effect