We have stayed at both several times (two of our most favorite resorts). For a first trip, I would stay at Wilderness Lodge. It is wonderful, and the proximity to MK and boat option will be a blast. Our first deluxe stay was at WL, and I still remember how special it was. The decor of both is similar, dark, and both have a bunk bed option, which is always a hit with our family.
I would double check with your
travel agent to see what room category she is comparing. I WOULDN'T stay at AKL without the savannah view, which usually bumps that up in price, making it more $$$ than WL.
Our next trip in Jan will be to AKL-Kidani. We have paid out of pocket for BLT (and LOVED it), and this trip we are renting points and staying with family in a 2BR. The pool at Kidani is probably my favorite between the two resorts. If you stay at AKL, you are allowed to use both pools, Jambo and Kidani. There is a bit of a walk between the two, but nothing a young family and eager kids couldn't handle.
That being said, there is a special feeling swimming at the WL main pool, something about the beautiful resort wrapping around the pool area that just made it seem special. We actually had some really fun encounters with wildlife during our stays. One time there was a mom with 8 ducklings that swam in the pool everyday (think they don't permit this anymore), and one trip we had a ground level courtyard (booked woods view), and every morning, ducks, rabbits, and something else would come and join us for breakfast on our patio.
The kids would also really enjoy Hoop De Doo show, which is a quick boatride from WL.
Good luck, they are both resorts...but definitely check room category comparisons as that might make your decision easier!