Since I'm seeing these questions start to pop up, I figured I'd share my 2 cents and let others chime in. Writing as the parent of a 5th grader who has been at the AAP center at Belvedere (as well as a newly admitted 2nd grader). Belvedere is also our base school, so this wasn't an issue for us.
Center and school-based level IV use the same curriculum.
Classes at center are only AAP eligible students. Classes in school based program are a mix of AAP eligible and other students -- ratio depends on how many AAP students elect to stay in the base school. A class that is all AAP is likely to move slightly faster and may be a better social mix for geeky kids.
Typically, centers have more AAP students, which allows for more extra curriculars, and more mixing kids up from year to year. But this is not necessarily true -- Belvedere has had only 1 or 2 classes per grade in the AAP center in recent years. When they have more kids than can fit in 1 class, but not enough for 2, they've sometimes gone to mixed grade classes. This worked out fine for my son -- especially since he had a really fabulous teacher when he was in a mixed grade class -- but I know some of the other parents were unhappy.
The board has considered cutting busing to AAP centers as a budgetary savings for several years, but has not done so.
If you have a younger child and there is room at the school with the AAP center, they will allow pupil placement of the younger at the center school in order to keep the kids in the same school. But, if your center kid graduates, the younger kid will have to go back to their base school (unless they are the right age and qualify for center services).
It is worth looking at the pyramids before making a decision -- in some cases, centers feed into different middle and high schools than the base school.