We have one AP Associate role with the Accounting Center. However, when one of our users logs in to the AP Associate role, they see a different center/menu than any other user in that same role. Any ideas why this would occur? We don’t use global permissions.

there's a user preference under set preferences where you can choose the classic center even if the role wouldn't normally have it. so one of those users probably has that set, and they can change it back if they want.