The two stories take place in the same historical time. Faith takes place in the last part of Empress Ki, after she gets married.
While in Goryeo reigned King Gongmin and Queen Noguk (Faith), in Yuan did Emperor Huizong and Empress Ki (Empress Ki).
The two states were in constant conflict with each other. In Faith, King Gongmin had to fight against those who wanted Goryeo to be another province of Yuan. And one of those, was the main antagonist, Ki Cheol, brother of Empress Ki.
Both are historical, action and romance dramas. Only that Faith it's a lot lighter, have fantasy and medical themes, and focused in a romance story between two characters. Empress Ki is a darker and more serious story, with lots of tragedies and political themes, that focuses on a longer range of time, and in which the romance is just another one of the many points that has to offered.