I don’t remember Grover ever being on the Muppet Show. Big Bird had a guest appearance once, perhaps that’s what you’re thinking.

Kermit was very much Jim Henson’s avatar and was the core character that the Muppet Show was built around, and existed before Sesame Street (IIRC, he was the very first puppet that Henson made, cut from his mother’s green coat). I believe that the Sesame Street Muppets other than Kermit are all owned by Children’s Television Workshop which is at least in part why they only made limited appearances in non-Sesame Street productions. There was a significant overlap in the puppeteering staff between Sesame Street and the Muppet Show (most notably Jim Henson and Frank Oz who were arguably the greatest comedy duo of the late twentieth century, but I think some other SS puppeteers also worked on the Muppet Show).

Of course, all of this is going from memory and someone going to Wikipedia could probably find all the places where I’ve misremembered things or am just plain wrong.

Grover too? To me, they're almost entirely separate. Kermit is the only one I remember who regularly appeared in both. I think there's a very brief Big Bird cameo in the Muppet Movie, but I cannot recall any other significant character from one show appearing as more than a brief cameo in the other.