>>718260125
To clarify >>718260592 point, Azula and Zuko acknowledge on some level that they should be family. But Zuko is Ursa's son and Azula is Ozai's daughter. Ozai was a genuine psycho who believed in might makes right and so Azula only respects strength. To the point she can't even function with normal people her age. So when she encounters Zuko, who's biggest character trait is his empathy, even if that empathy was poisoned by his father for a while, Azula only sees a weak toy to play with. A favored toy but still a toy.

Azula is kind of a fucked up character in general. I don't think she could ever be saved, she's too far gone for that to happen. But she is a weapon Ozai manufactured. As far as Azula knows, she has the literal divine right to rule and fear is her weapon of correction. So at the end of the series, when those weapons stop working, she just lashes out at everything. And when she's beaten, the only thing she can do is cry as she's haunted by the image of her mother.