The advantage of Homebrew is that it is modern, small, and simple. Obviously apt is better, but apt also requires that you maintain a large repository of binaries. Homebrew is slow[1], but because it builds from source it is trivial to create a new package. Homebrew packages are just Ruby files. If the project uses Autotools, all you generally need to do is enter the URL, an md5, and the dependencies, then it Just Works. If it uses waf or something it's still just a simple modification.

Instead of reinventing its own selfupdate functionality, Homebrew just uses git. Updating Homebrew is really just fast forwarding a git repository. Finally, the maintainers explicitly do not duplicate the libraries/tools that ship with OS X.

This is a big step up on OS X from Macports, which installs itself in /opt/local and rebuilds everything possible. Homebrew is instantly understandable. Packages are files, you just match them by name. apt is admittedly more complex, but at least you get something for that complexity.

[1] really what's slow is building from source. Homebrew itself is very fast.