Both dramas are about the female lead trying to make it as an actress by starting from the bottom. In both dramas, they get help from a veteran actor. In Skip Beat, the main girl uses her past to fuel her ambition, while in Dear Prince, the main girl struggles to overcome her phobia. Dear Prince is more of a supernatural drama, while Skip Beat can be considered an idol drama.
Even though the plots are different from each other, the main concept of a girl trying to change into something better is what links them together. In Skip Beat! the main character is going through a change of look/career/life out of revenge, while in Yamato Nadeshiko Shichi Henge, she is being transformed into a lady for the her own sake. Both dramas are funny, with romance aspirations and with moments that will make you "awwwwww". Not to mention the actors are really skilled. I really recommend both of them.