It's important to note that while Azula is falling, Zuko shows genuine regret as he believed she really was going to die. This, coupled with Azula's affection for Zuko during their time together proves that at best, they do care for one another, but due to their different views on life and the nature of their upbringing (Zuko through Ursa and Iroh, and Azula through Ozai), they cannot coexist. While initially vague, as the series progressed it became clearer where Azula's rage towards her brother comes from. She impulsively feels the need to berate and belittle him and rub their father's favoritism of her in his face, even when the situation doesn't call for it and after Zuko makes it clear he no longer cares for his father's approval. This apparently stems from a massive and subconscious jealousy of her mother favoring Zuko.