Avatar: The Last Airbender is famed not only for its stylish Asian-inspired setting, but also its rich character arcs, especially that of Prince Zuko of the Fire Nation. Zuko's heart is being tugged in different directions in the story, and he is torn between a superficial notion of honor, his desire to prove himself and his own inner goodness.
Zuko has had many influences, and without them, he would likely have turned out much worse. His short temper, thirst to prove himself and resentment toward his father could have led him to be much more like Admiral Zhao, who is basically Zuko without any of the positive influences. Zhao meets a bad end, and in an alternate timeline, Zuko could have gone the same way.
How Zuko & Zhao Overlap As Fire Nation Hunters
Prince Zuko and Admiral Zhao have a lot in common with one another, including their Fire Nation origins and positions of command. Admiral Zhao initially has much more authority than Zuko does, but Zuko still has his own ship, allowing him to demonstrate what sort of leader he is. A lot is expected of them both, and aggressive, ambitious characters like Zuko and Zhao will either crack under the strain or hone themselves into a fine edge.
Before Zhao's eventual downfall, Zuko and Zhao endure that pressure in similar ways. Both characters are short-tempered and result-oriented commanders who are quick to punish failure (or at least throw a tantrum about it). Both are also often foiled by their own anger and recklessness, such as when Zhao accidentally burns down his own fleet while attempting to capture Aang near Jeong Jeong's hideout. Zuko and Zhao resort to extreme methods to get what they want, from Zhao killing the moon spirit itself to Zuko risking his own life to capture Aang in the North Pole or freeing Aang from Zhao so he can have the honor of capturing Aang himself.
How Zuko Diverges From Zhao's Path
A few key factors save Zuko from becoming just like Admiral Zhao, including Zhao's sticky end. Zuko had the benefit of his caring mother, Ursa, who doted on him and brought out the best in him. Zuko's good side was buried, but not killed, when Ursa fled the Fire Nation capital for her hometown, leaving Zuko alone with his cruel sister and father. Ursa had already nurtured Zuko's good side, and later developments help him recover it, while it's unclear if Admiral Zhao ever had such a nurturing figure in his life.
Uncle Iroh, Ozai's brother, also brings out the best in Zuko, and urges him to abandon his pride and realize his true destiny. At first, Zuko and Zhao are both prideful Fire Nation commanders and Avatar hunters, and it is Zhao's excess pride that drives him to slay the moon as part of his Northern Water Tribe conquest and earn a distinguished place in history. However, the reverse occurs; Zhao is overwhelmed by the reborn moon spirit and history soon forgets him. Without the positive influences in Zuko's life such as Iroh and Ursa, Zuko would have similarly plunged the world into darkness to satiate his pride and honor. Zuko eventually abandons the trap that is ambition and pride, but Zhao never does, and he pays dearly for it.
Toward the end of the series, Zuko becomes a far humbler person and lets go of his mission to capture Aang for his father's sake. He accepts his role as the Avatar's helper and friend, allowing Aang to confront Ozai alone and save the world and helping Katara to take down Azula. Zuko also learns patience and flexibility from Iroh and Aang, as opposed to the impatient and greedy Zhao, who we learn abandoned his teacher, Jeong Jeong, before even completing his firebending training. Zhao arrogantly thought he had learned all he needed, while Zuko eventually learns true firebending from the only two surviving dragons.
Both fire and life are about more than destruction and glory; they're about what can be created and protected. Zhao was never ready to learn that lesson, and without Iroh, Aang and the dragons, Zuko would have likewise been swallowed up by his own flames.