Rewatching the series. This scene was always weird to me as it was odd that Ozai would toss away one of his biggest weapons. But now I know why he did not let her come with him at this moment. After Zuko told his father than Azula lied to him, he saw her as a schemer who may try to backstab him like he did his father for the throne. She was too much like him to the point that now she was a burden like Zuko. Ah, it makes so much damn sense now.