Nothing so dramatic.
Prior to Sesame Street's creation, Jim Henson (creator of the Muppets) already had a show called The Muppets. The cast of this show are the Muppets of The Muppets franchise (including the movies).
When The Children's Television Workshop started developing Sesame Street they wanted puppets to be part of the program, so they asked Jim Henson to provide these puppets. So Jim Henson developed a separate cast of MuppitsMuppets for Sesame Street, but also added his signature character from the original Muppets: Kermit the Frog.
That's why the two franchises have different characters, and thus different rights.