The biggest advantage is when removing packages; its dependency resolution is (supposedly) much better so it's able to find and remove orphan packages more accurately without hosing your system.
It's also got a much friendlier set of commands:
$ apt-get upgrade vim # aptitude upgrade vim
$ apt-get dist-upgrade # aptitude full-upgrade
$ apt-cache search # aptitude search
$ apt-get install --reinstall vim # aptitude reinstall vim
$ apt-get remove --purge vim # aptitude purge vim
The list goes on.
It's also got an awesome ncurses based interactive interface if you just type:
$ aptitude
Great for sysadmins who miss synaptic or anyone who feels like a GUI is just too mainstream.
apt-get also is able to remove orphaned libraries.
Among other "advantages" of aptitude, it can be terribly slower than apt-get.
And it has a "smart" dependency engine that often proposes convoluted solutions (involving uninstalling packages I don't want uninstalled) when apt-get will simply update the dependency and their dependents... But at least, if you don't agree with the first solutions, it has others (sometimes including the one apt-get chooses).
Aptitude has an optional screen-oriented GUI, much better search and filtering capabilities, a listing of "new" packages each time you update, and an excellent dependency resolver.