It's really just a higher level of service. Basically, the concierge rooms are the same size as the standard rooms, views are usually the same and most of the basic room amenities are the same. In concierge though, the rooms are either located on a private floor or in a private building only accessible by using your room key.
Prior to your trip, you get the assistance of a pre-planning team. They'll make all your ADR's, arrange tickets, transportation, tours, special celebrations, purchase gifts ahead of time, setup your room for Christmas, etc. This is usually done via e-mail and even though they have to wait until the 180 day mark, you can send in your requests well ahead of time and have them logged in. Some of the resorts will send you a nice pre-planning packet to help you out. It's fairly common that concierge can arrange for hard to get reservations, like a princess breakfast or Illuminations cruise. The restaurants and tours do not "set aside" tables or seats for concierge guests, but because the staff is on the computer most of the day, they can quickly and easily check for cancellations.
With the exception of a few resorts, when you arrive, you are met by a concierge greeter and escorted to a private check-in area. Staff are available 7 days a week from 7am-10pm to assist you with any last minute arrangements that you might need. Lounges are also available during that time with different offerings usually tailored to the theme of the resort. For example, at the AKL and GF, they offer an afternoon tea. In the evenings, wine, beer and cordials are available. Other than breakfast, the food is not really designed to take the place of an actual meal, but it's a nice supplement throughout the day (if you are going to be around the resort). Coffee, tea, hot chocolate and cold drinks are also available.
As Marimar mentioned, the AKL offers a sunrise and sunset safari exclusively for their concierge guests and there is also a Wildlife Safari Excursion that is available for booking by any concierge guest.