I think Targaryens are showing their true nature when they face with the tragic incidents.
I would say for King Aerys is that
- He felt incompetent when people say his hand Tywin Lannister is successfully ruling the 7 kingdoms and not Aerys and not only that Tywin was married with Joanna, a woman that Aerys wanted,
- Secondly he lost a son and then executed the wet nurse for that and tortured people for that,
- And lastly he has been captured by the Lords of Duskandale and imprisoned for 6 months, I think after that he totally gone mad and his Targaryen coin showed its true face.
The snapping points for Daenarys were;
- The failure of Tyrion, when he was outwitted by the Lannisters, they lost the allies from the Iron Islands, Dorne and the Highgarden, and she blamed Tyrion only and she was enraged, and she even blamed him for protecting his family.
- Losing Viserion to the Night King and army of the dead, it must be a big shock to her to see such a supernatural threat in Westeros
- Occurrence of Jon's secret identity as a Targaryen, to learn that she isn't the only Targaryen and there is someone with better claim than hers after so many things she did for the Iron Throne
- Then again, losing her best friend and another Dragon, and then she executes Varys as well.
These incident pushed her to lose control, and she basically turned into another mad Targaryen, even Jon, someone she loves according to her, failed to convince her to spare her enemies lives and she didn't want to show any mercy and she wasn't going to show any mercy in the future, and she would continue to burn her enemies like mad King did, the only difference is that Aerys had pyromancers, she had a Dragon.