The books and show diverge as regards Jaime's relationship with his family. In show and books, Tywin wants Jaime to leave the Kingsguard (with the High Septon releasing him from his vow) and become Lord of Casterly Rock. Jaime refuses and Tywin is angry, saying he no longer considers him a Lannister.
But in the show, Jaime later offers to Tywin to leave the Kingsguard and become Lord of Casterly Rock if Tywin will spare Tyrion's life after his trial. Jaime and Tyrion later share a hug goodbye, it's all very touching.
In the books, Jaime never reconciles with Tywin, and he arranges Tyrion's escape through Varys. During the escape, Jaime tells Tyrion that his wife from his youth, Tysha, was not really a whore arranged by his father to teach him a lesson, she was a real woman who really had wanted to marry him. Tyrion is furious and, partly as a result of this, falsely tells Jaime that he killed Joffrey:
Quote:
“You poor stupid blind crippled fool . . . I am the monster they all say I am. Yes, I killed your vile son.” He made himself grin. It must have been a hideous site to see, there in the torchlight gloom.
So in the books, Jaime is even more disillusioned with his family.