I was still watching it a lot then (I was 6 to 10 years old). I don't remember feeling like anything was changing except around 1992 (even before the theme music change) I started to notice that the special effects were getting more elaborate than they used to be. Like Elmo imagining he's flying, being superimposed against a blue sky. Simple stuff, really, but more elaborate than anything they had really done before that I was aware of. I remember thinking that it was an improvement.
I have six taped episodes from 1991 and 1992. To my mind, it was still pretty much the old show. A lot of old material was still floating around, and the tone was pretty much as it had been in the 80s.
One possible difference: I think maybe the early 90s was when the loose, improv-like dialogue among the actors began to be simplified, and the characters stopped bickering the way they used to all the time. Not sure---that's just the impression I get comparing these '91/'92 eps to ones from just a couple of years earlier.
The show then was a very schizo thing, really---one foot in the 70s and another foot in the (then) present---and kind of unsure where it was going.