Both are the stories about how a woman (a princess in QSD and an aristocrat daughter in TEOC) becomes - after facing many obstacles - the ruler of a country. Both are based on real characters, however their stories are dramatised for the shows.

The main difference is that while in QSD the heroine climbs the ranks first as a cross-dressed boy and then as a princess, in TEOC, the main character has to climb through the emperor's harem (and so the story, especially the first part, is more heavily focused on harem intrigues and conflicts), which is then replaced with more political plots.

Both have tragedy as well as a fair dose of angst (more so for TEOC though), well developed characters, are exciting and completely suck you in.

After finishing both there's this feeling of melancholic bitterness.