The main reason we booked a Concierge stay was so that we could try the Sunrise Safari which is only available to AKL/AKV Concierge guests. It was $55/adult when we did it (I think it has gone up a bit since then) and includes breakfast. It was a great experience.
We ended up enjoying the other Concierge amenities more than I expected including just hanging out in the lounge, especially since we had booked a studio that trip. It was like having a large, shared living room overlooking the beautiful AKL lobby. The food offerings in the evening were great even though as vegetarians there were only a few things we could eat. The desserts were wonderful as were the cookies in the afternoon, though I wasn't too happy when I stepped on the scale when I got home from that trip!
The only downside is the view from those rooms. We did get a savanna view, but you are up so high that the tree tops block part of the view, especially if you are in the room furthest from the lounge. There were times when there were animals close to our room but we couldn't see them through the trees. As a result, we didn't spend nearly as much time on the balcony as we usually do. I'm glad we tried it and we will probably try to book it again some day, especially on a trip where we plan to hang out at the resort.