Depends on what you mean by "perks".
It's usually a key access floor or area. There's food available in the lounge and turndown service nightly. And CM's of varying quality who will make reservations or hand you brochures. I've had friendly CMs and some that bordered on indifferent. It just depends on who is working that day. They don't really pull strings for you or anything. If the restaurant is booked, they tell you "Sorry, can I check something else?". At least, that's been my experience.
AKL concierge will allow you access to the Sunrise/Sunset safari (you still pay, but only concierge guests are allowed to book it).
Sometimes check-in is faster, but just as often it hasn't been. They do deliver your packages to the concierge desk or sometimes to your room. That's kind of nice.
But honestly, there isn't much draw for concierge besides the food. It's nice, and sometimes even a savings, if you can score a cheap upgrade, but nothing I'd pay a whole lot more for.