The Music
Mickey's Christmas Carol has a delightful score by Irwin Kostal, which does a lovely job mixing classic Victorian-era holiday tunes with a modern take for the saga. There aren't original songs, but the chorus for some classic Christmas tunes fits well and can move from bright and cheerful to sinister to boisterous at the end. It's not top-tier Disney music, but it does its job.
The Muppets, on the other hand, are bursting with songs, several of which are very good. Kermit gets the great "One More Sleep Til Christmas," Christmas Present gets "It Feels Like Christmas" and more. They don't all land as well, yet they carry a wonderful spirit. The big one is "When Love is Gone," deleted from the theatrical release but thankfully reinserted for home video. This is the core of the movie, the moment Scrooge chooses greed over love, and the reminder that he made the wrong choice. That song alone is one of the best in the Disney or Muppet canon to elevate this movie.
WINNER: Muppets
The Cameos
Each movie does integrate characters into the story in different ways. The Disney version is notable for lesser-known characters like Toad as Fezziwig, Ratty and Moley as collectors and the Three Little Pigs as carolers. They don't dominate too much, are short and fun, and have a nice backdrop to things.
The Muppets make a mistake in trying too hard to fit as many Muppets in. Some are good, like Fozzie as Fezziwig and the Electric Mayhem as the band. But others are distracting such as Bunsen and Beaker as the collectors, Bean Bunny as the little kid Scrooge talks to and the Swedish Chef cooking. There's also how the movie basically stops just to spotlight them. Disney was a bit better letting the cameos be brief while the Muppets goes too far spotlighting so many of their guys.
WINNER: Mickey