There is an emotional connection/ bonding process central to the plot. The main characters in both have large overlaps in their age, status, and experiences.
Both feature a male lead with split personality (Kill Me Heal Me has several personalities while Gap Dong only has two) and an unpredictable plot based around mystery. However Kill Me Heal Me is for the most part more light-hearted and comedic than the rather heavy storyline of Gap Dong.