Mise vs Homebrew
I use both Mise and Homebrew in my daily professional and personal work. There are some overlap between the two. I used to just use Homebrew for everything until I encounter incompatible tool versions from ever growing simultaneous number of projects I'm working concurrently.
When to use Mise
Mise really shines in managing programming language versions. Different software projects tend to use or stick different version of a language. It switches seamlessly when going into or leaving the project directory. Once activated, everything that is related to the language like dependencies are handled and stored by mise in it's own directory.
One other feature of mise that became my favorite these past few days is setting the environment variable only inside the project directory. Pretty much what Direnv is doing but is already integrated in mise. Having one less tool to deal with is always a good thing.
When to use Homebrew
Homebrew is really good at installing tools and applications where you don't need to have multiple versions of it. You, generally, just upgrade the current installed one to stay current.
There will be cases when I also install a programming language through Homebrew. This is when the language needs a tight integration with the OS or another Homebrew install requires a language to be installed first.