'Castigate' indicates a very strong level of criticism that is more intense than simply suggesting improvement. The director castigated the actor for repeatedly forgetting his lines.
Pro Tip 2/3
Emotional Weight
Using 'castigate' can carry an emotional impact, suggesting disappointment or disapproval from the speaker. The community castigated the council for ignoring environmental concerns.
Pro Tip 3/3
Public vs Private
'Castigate' often implies that the criticism happens openly, in a public setting, not privately. The official was publicly castigated for the scandal.