Lora,
The answer is, of course, both!
No, seriously, both are great experiences, and it is all about what you want to do. Here are some comparison points;
Spirit of Aloha
Pros
Wonderful Luau ambiance
Decent food, with some mildly-exotic choices, but plenty of kid friendly foods
Fun, warm experience that allows for both family time, but also a show
Kids often get to dance with the entertainers at the end of the show
Covered seating, but cancellations can happen with bad weather
Cons
No Disney characters
Mickey's Backyard BBQ
Pros
Wonderful outdoor, casual setting
Decent food with a focus on (duh!) barbecue
Fun experience with good live music that sets the mood
Completely outdoors, so not the best at bad weather times, and can be cancelled due to bad weather
Cons
While kids can dance with the characters, there is no autograph time.
So which would I do. Hard to say, but with a 6 and 4 year old I might lean slightly to Mickey's Backyard BBQ because it has both a character experience and live music. But either is great!
I am glad to answer any follow-up questions!
Have a magical (and delicious) vacation!