A machine dedicated to being a server doesn't have to run the game. More bandwidth is almost always available, as users at home are mainly on cable/dsl (Some people aren't able to disable their wireless modems aswell, which is horrible for gaming). With complete power and bandwidth, more users are allowed to enter the game(See World of WarCraft's 40vs40 or just Realm/Server population). Unlike home computers/consoles, a super computer could be used as the server.
Dedicated servers are almost always online. Players will flock to a certain server and usually make them their gaming homes. Because of dedication, you don't have to fear showing users what their ping/latency is. Admin rights grant users power over their servers. Get rid of annoying users, team killers, change levels/rules at a whim. Because of more bandwidth, it's easy to dl custom maps/others features.
The biggest downfall is somebody has to pay for the server. World of WarCraft has users pay Blizzard, which has led to incredible profit. Valve/id/other FPS'ers usually just have users buy their own servers from server companies. I believe id with Quake Live now has you buy their servers, which in return gives them a bit of profit (:
The next big step in online gaming, banning users with above 30/40 ping :d
// Curtis Turner <IceIYIaN>
// Creator of Elements of War!
// http://www.ElementsofWar.NET
// http://www.ModDB.com/Mods/Elements-of-War