Tyrion's story about Tysha is similar to its version in the novels, with several changes: according to the novels, Tyrion was thirteen at that time, not sixteen; Tysha was the orphaned daughter of crofter, not wheelwright's; after Tysha was gang-raped by the guards while Tyrion was forced to watch, his father forced him to be the last to take her.
Maester Aemon says that his father was King Maekar Targaryen, who was followed by Aemon's brother Aegon, who was then followed by Aerys "the Mad King" Targaryen. In the novels, however, Aerys was the son of Jaehaerys, Aegon's second son, and Aerys was Aemon's great-nephew.
The trivia items below may give away important plot points.
In the novel "A Storm of Swords", Jaime reveals to Tyrion the truth about Tysha: she was not a whore, but what she seemed to be - a crofter's daughter whom they encountered by chance and saved her from rape. Tywin was angry when he found out that Tyrion married a lowborn, and decided to teach him a sharp lesson: he commanded Jaime to falsely tell Tyrion that Tysha was a whore who had been paid to make him a man. Jaime felt bad about that, but did not dare to defy his father. When Tyrion learned the truth about Tysha after so many years, bearing in mind how she was gang-raped by the guards and by himself at Tywin's command, he furiously vowed to exact revenge against Jaime, Cersei and their father.
It was revealed in an interview in March of 2018 with Sean Bean, that when Ned died, he whispered a small prayer to himself. In the beheading scene, he is mouthing some words, which up until then were a mystery.