I am a former WDW concierge and the site given above is a good explanation.
A Disney concierge kind of serves a your personal assistant. We start weeks before you arrive, contacting guest, taking care of reservations, tickets, special occassions etc. The cms on the concierge levels are restricted to those who are staying in concierge rooms and they go the extra mile to make your stay special. They are very knowledgeable not just on Disney but the surrounding area. They required us to know a specific number of guest names on sight.
To me the best part of staying on the Concierge level is the constant snack bars starting with continental breakfast, tea, cocktails and afternoon/evening snacks. We also had special parties for our guest for events like the superbowl and halloween to name a few.