The Watchers on the Wall - S4-E9
Revealing mistake: Castle Black defenders drop heavy stones on the free folk and one of them gets hit on the head but forgets to act injured (or dead). (00:23:20)
The Mountain and the Viper - S4-E8
Continuity mistake: When Oberyn first tells Gregor that he will make him confess to raping and killing his sister Elia and her children, he is holding his spear with both hands. In the next shot, he's holding it with just one hand.
Two Swords - S4-E1
Continuity mistake: When Arya and Hound come upon the murdered people Arya is riding facing left. After they pass, she is facing right. (00:48:40)
Answer: Basically it's because he broke his vow. A member of the Kingsguard is sworn to protect the king at all costs. Jaime elaborates more on the deed to both Catelyn Stark and Brienne of Tarth, telling Catelyn that no matter what course of action he took, he would be breaking one vow or another (i.e. if he obeyed the king, he would conversely be disobeying his own father), and telling Brienne that the Mad King was planning on burning all of King's Landing, but he did not bother to tell Ned Stark that. Ned Stark felt that killing Aerys was dishonorable and excessive. Robert Baratheon still could have usurped the throne without Aerys being killed.