Both exist. Fitting rooms are in department stores; dressing rooms are in theatres.
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MUSCOVITEfitting room is an area in a store where you can put on clothes to see how they look ... and this and only this term is used on either side of the Atlantic to express this idea?I wouldn't call it a dressing room, that's all I can say. In my AmE, a dressing room is backstage near the green room.