mise has more features - it is a super set of asdf. For example it can set your env vars when you cd into a directory (like direnv). It also has tasks (which I haven't used) - they appear to be similar to what a Makefile does. So you can potentially replace three tools (asdf, direnv, make) with one.

Mise started out using the same plugins as asdf, mostly focused on adding performance and usability improvements. Over time it added more features and security.

Most tools are now directly fetched from github releases without the need for random shell scripts (which is what asdf plugins are).

It also grew to be a task runner and environment manager. At first you might think this is scope creep but they're both opt in and very elegant additions. I don't want to ramble but let's just say they've solved real problems I've had.

I'm a fan of it, and I can't think of a reason why I would use asdf over mise. Its real competition is nix (+devbox/devenv/flox), devcontainers, and pixi.