Act, play and impersonate all are synonyms when they mean to assume the appearance or role of another person or character.
Act usually and play frequently imply feigning for theatrical representation
- act Hamlet
- play the melancholy Dane
Even the idiom “to play one’s part” has a theatrical origin and still connotes performance and a contribution to an ensemble.
Whether impersonate implies simulation for the sake of theatrical representation or for deception can be gathered only from the context.
- an actor who impersonates women
- he was arrested for impersonating an officer