Well, in general, they can have different sound qualities. Tube preamps have the tube sound, and solid state preamps may have a clinical or warm quality, depending on the design and implementation. These differences can be very subtle, and they may not actually be worth that much. I suppose that a "perfect" preamp will impart none of its own sound to the end result, but that's rarely the case; it is manipulating the signal in a variety of ways, after all.
Some preamps may have a phono input, and some may not.
Preamps with digital inputs will have DACs. The DAC used, as well as the implentation of surrounding electronics can again generate a different sound.
If you are talking about home theater preamps, then there are myriad features that can be considered -- does it support all audio formats, does it support audio via HDMI, how does Zone 2 work, is bass management implemented the way that it should be, do you like the remote? Those are just a few off the top of my head.
Unless you are working with esoteric tube gear, I don't think there's any need to match your preamp and amp (I don't have much personal experience with tube gear, but I understand it can sometimes have impedance matching funkyness). As long as you like the sound and feature set, you should be good to go.
Anyway, I'm sure there are more things to consider, but that's a start.