So, I think that Azula and Zuko tell the story of what happened to Ozai and Iroh, but on like x2 speed

Think about it. Iroh was the first born, and a feared General, until his military failure that was also tied into the loss of his son.

Zuko was the first born, and was at a military meeting when he had a failure that caused him a great loss, his family.

Que soul searching quest, that involves finding yourself, realizing that the Fire Nation is a lot wrong about a lot of different things, finding the last two dragons, learning about the other nations, learning about balance, and making friends with a LOT of people that later turn out to be completely OP.

(Ok, Ozai always hated Zuko. We have no evidence from the show that Azulon didn't also hate Iroh.)

Azula is the younger child, pushed to perfection, pushed to do something to harm her brother in order to gain the throne/her father's approval/usurp her brother's rightful place as heir. (Hunting him down, returning him to the Fire Nation as a prisoner, I mean, which example do you want?)

Ozai was also the younger child, pushed to do something to harm his brother to gain the throne. (In this case, it was kill his son, but he knew Iroh looked at Zuko as a son instead of a nephew.)

Anyway, that's my theory, I haven't read the search or anything following the series, I just thought it interesting that they parallel.