I'm very unsure of my ability to be analytical about the first three Muppet movies, because my brother and I watched and rewatched them so obsessively as kids. The trilogy (which is in no way a trilogy) of The Muppet Movie, The Great Muppet Caper, and The Muppets Take Manhattan are my first point of reference for what the Muppets are and ought to be, and it's very hard to break out of the "OMG CHILDHOOD JOY AND LOVE" thing and see them on their own as movies rather than beloved cult objects.
If I were breaking it down, though, I'd have to say that:
1. The Muppet Movie: has the strongest emotional content, the most to say artistically and philosophically, some of the best visuals, hands down the best music, and is essentially a love letter to America and at least the idea of Hollywood
2. The Great Muppet Caper: has the best story overall (bearing in mind that plot cohesiveness isn't really ever the Muppets' strong suit), the best guest stars and use of cameos, some of the best technical work (the multi-bicycle ride number! HOW IS THAT POSSIBLE?!), is generally the funniest, and is essentially a love letter to England
3. The Muppets Take Manhattan: has some of the best set pieces, gags, and comedy sequences, interspersed with some stuff that doesn't work 100%, has some moments of genuine emotional power (I have trouble not wibbling at "Saying Goodbye," and the marriage scene at the end always makes me weep. Don't judge me), though they're not quite as fresh and authentic as in TMM, some fun meta stuff that I still find relevant in my daily life ("Ocean Breeze soap will get you clean!" "You mean we should just...say what it does?!"), and is essentially a love letter to Broadway.
I agree that the musical at the end of TMTM is exactly as great as it's been built up as, which is pretty amazing. This was one of my gripes with the most recent movie, which I...mostly enjoyed. But there was all of this build up about how awesome and inspirational the original flavor Muppets were, but very little display of their actual awesomeness compared to the amount of time spent showing them being sad and remorseful. :/