These are not accurate in every situation, but are rather generalizations.
Sealed subs generally have better transient response, which is often attributes to their musicality. They have a gradual rolloff as the frequency goes down. They generally allow a bigger margin for error during the actual construction, and are simple to design and build. They generally have a higher power handling.
Ported subs can generally go a bit deeper depending on the frequency to which its been tuned. Rather than a gradual rolloff, a good driver will be fairly linear to the tuned frequency, and then drop like a rock. The margin for error in the construction is smaller than in a sealed.
Both designs have their tradeoffs, but when properly designed, you should have a good end result either way. A lot of it depends on the driver itself, and its paramters, particularly the q of the driver. I personally use a ported sonosub tuned to 16 hz and love it, but I've also heard a lot of sealed subs that are tough to top...