The differences are that homebrew installs to /usr/local by default, doesn't keep track of package versions, and doesn't install its own versions of libraries that come with OS X (and therefore might be screwed up by an OS update). These are all Really Bad Ideas, which make homebrew technically inferior to macports, and less stable. However, they also make homebrew really simple, so it's much quicker and less cumbersome to use. Almost the definition of quick-and-dirty.
Homebrew does not use sudo, installs everything to /usr/local (which keeps major OS X updates from destroying your dev environment), and it's much easier to edit and contribute recipes back to the main project (for me, anyway).
Homebrew has far better dependency management then Macports. I fine that using Macports you have to download and compile the kitchen sink every time you want a moderately complex package.
Homebrew will use libraries that are already present on your system, instead of keeping a separate copy of everything.<p>The downside may be decreased stability, but the reduced compilation times make up for that, to me (and haven't had a problem yet).