Both dramas have female rivalry in theater, one a naturally gifted amateur and the other a rich, well trained actress, who compete for the lead roles in their productions with a strict matriarchal mentor. Both are based off of comics.
Watashitachi wa Douka Shiteiru and Burn Down The House are very similar in plot and themes. Family secrets. Revenge. Hidden Identity. At the center are two female leads fighting to prove their mothers' innocence. Burn Down The House is a better made and more engaging story, but Watashitachi wa Douka Shiteiru leans more towards romance with traditional wardrobe, settings, and food that gives it an interesting spin.
'That Winter, The Wind Blows' is a 2013 remake of this japanese drama.