I have stayed at both and that is a VERY hard choice to make.
If you are comparing only the hotel itself, I would go with AKL.
If you are comparing all benefits of both resorts, I would say BC.
We absolutely LOVED AKL when we stayed there. When we were planning the next trip, we booked BC but I was very much on the fence about leaving a resort that I loved.
The BC did NOT disapoint. The proximity to Epcot & DHS, the many options for transportation (or walking), the dining options, the entertainment and travel times all gave the BC a "one-up" in my view.
We are returning to the BC for our upcoming trip (thanks to the 7 for 4 promotion) if that says anything.
AKL is a beautiful resort and has a lot to offer. For our family though, everything else that BC offers makes it our new favorite.
Good luck with your decision! I vote for staying at them both and seeing which your family prefers.