Hummm... both are the same.That's how it works in practice, but the thing is general servers could be utilized for multiple games.
The difference is what you are doing on server... for a general server you possible only does authentication and mp user list while in a dedicated you do almost everything... both are a hardware in some place talking resources and they should be shout down after a time or lower usage.
If no one is playing a game, instead of it getting shut down, those resources get used by a different game. And someone comes back to play that game, it utilizes some open space of the server.
Similar to how a regular PC works for everyone else in the world. You don't have to restart your PC if you are done using Google chrome and want to play a game. The computer simply turns off the instance of Google chrome, and starts an instance of the game.