While faintly sympathetic toward Zuko, Azula impulsively deems him pathetic, albeit in a somewhat playful tone. Their attitude toward one another is unconventional, solely due to their different views on life and the nature of their upbringing (Zuko through Ursa and Iroh, and Azula through Ozai). In this manner, they cannot comprehensively coexist.
It is important to note that while Azula is, in many instances, determined to destroy Zuko, she shows apparent genuine emotion toward him. In "The Beach", Azula tells Zuko to leave their beach house, showing that she understood the feelings Zuko had in connection with their previous home. More so, Azula wanted to ensure that Zuko was happy, by allowing him to pursue a relationship with Mai, among many others. However, these signs of affection were only apparent when Zuko was showing loyalty toward Azula and his father.
While initially vague, the origin of Azula's rage toward her brother became clear. She impulsively feels the need to 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.