You’ll need a PhD for a career in primatology, and you can get into a good PhD program from either school. You’ll have to decide whether you prefer the more undergraduate-centric education at App or the wider variety of courses at NC State.
I’m a bit biased (I’ve taught field biology at App), but App is a great choice for an undergrad studying organismal biology. App has relatively few master’s students and no PhD students in biology, which means undergrads get most of the faculty attention. There’s plenty of good opportunities for field research since the NC mountains have so much biodiversity.
NC State has a much broader array of majors and courses in the biological sciences, but it also has many more undergraduate biology majors and MS/PhD students. State’s zoology program is exceptionally strong, so if you’re sure that’s what you want to study, there’s arguably nowhere better for a NC resident.
As a student at NC State, you’d be relatively close to the Duke primate center, which offers undergraduate research/volunteer opportunities. Duke’s primate center is one of only a handful in the US, and it’s unique in focusing on primate ecology rather than medicine. Additionally, NC State students are allowed to take courses at Duke, though very few take advantage of the consortium since it’s much easier to travel between Duke and UNC than between Duke and NC State.