Use castigate when you mean reprimand but in an especially harsh way. If you take a mean teacher's books, even accidentally, you might worry that she's going to castigate you as soon as she finds out.
Castigate means punish, and punish harshly, but the punishment is always a severe scolding. Sometimes it means criticize severely. Politicians in the Senate are always castigating each other for their alliances and opinions. Castigate and chasten, which also means "to reprimand" but is less severe, share the Latin root castus which means "pure." Ideally, if you castigate someone, you mean to guide someone away from the wrong path and toward a more pure one. But it sure doesn't feel like that when you're being castigated!