Both series involve a young, naive, and silly girl forced into a marriage with a serious, cold guy. She then has to learn to behave in a certain way as she is thrust into the public spotlight. In Goong, the guy is the crown prince while in Prime Minister and I he is the Prime Minister.
Both involve a young woman which's life suddently turns in such a dirrection that she has to take care of few children, which aren't her own. She has to lie about her life and live into a rich man's house. In those dramas the male leads are somewhat agressive towards the female lead as they suspect her honestly loving them (which was the case at the begging).