Katara and Azula orbit each other constantly as rivals, though they tend to avoid contact as Azula would prefer to antagonize her brother and the Avatar himself (or perhaps because she knows her chances fighting Katara, a bender just as prodigious as she is, are much slimmer). Their lives and narrative arcs run parallel to each other, two fourteen year old girls with complicated parental relationships who were forced to mature at far too young an age. Their traumas and experiences neatly reflect each other, positioning them as perfect literary foils. It seems fitting then that the final showdown between Zuko and Azula actually ends with Katara defeating and subduing Azula, chaining her down to a drainage grate using her waterbending as Azula screeches blue flames.
They have similar family dynamics and traumas, both losing their mothers when they were young in different ways, both having older brothers of the same age, and both have incredibly complicated relationships with their fathers. Where Azula lost her mother first emotionally and then literally once she was banished (both the fault of Ozai), Katara lost her mother when she sacrificed herself to save Katara and protect her from the fire nation. She is burdened by the knowledge that her mother sacrificed her life to protect her, while Azula is burdened by the knowledge that her mother rejected Azula in some ways because of how she resembled her father. These losses function to mature them both differently. The absence of Ursa means Azula is molded beneath the abusive hand of her father, who disregards empathy and softness. She becomes unhinged at his manipulation, obsessed with proving her worth as a firebender and her harshness as a leader because those are the things he values. Azula is never shown the value of compassion. The empathy she might have potentially received from Ursa is something she never experiences because of her banishment and she is forced to mature far too fast without any kind of humanity.
Katara is forced to mature because of the sacrifice Kya made, as well as the absence of her father. Because Kya abandons her out of sacrifice, Katara is shown by example what the most extreme form of love, empathy, and care is, which turns her into someone whose self-worth is invested in how she can protect and take care of others. She matures in this way by essentially taking on the role of her mother, becoming overly responsible and emotionally supportive for Sokka, Aang, Toph, and everyone else, who are all just as much children as she is. Katara looked up to her mother more than anyone, so the way that her mother leaves her encourages her to emulate her compassionate and caring nature, which is the opposite of how Ursa being forced to leave functioned for Azula.
This influences both of their relationships with their brothers. Azula and Zuko are complete rivals, constantly taunting and antagonizing each other, whereas Katara and Sokka have a relationship based entirely upon unconditional support and love. Furthermore, Azula sees Zuko as an extension of her mother, who made apparent her favoritism toward Zuko, which makes her hatred for him even stronger. On the other hand, Katara sees Sokka as a positive connection to both of their parents, which only strengthens her love for him.
In terms of their power as benders and fighters, they are relatively equally matched. Both are prodigies of their own sort, possessing rare prowess and gifts for even rarer substyles, Azula with her lightning and blue fire and Katara with her healing and bloodbending. They are each intimidatingly skilled in their own right, but both also deal with similar amounts of sexism. Azula is often categorized as the fire nation princess, a title always spoken with a certain degree of femininity that seems to cast aside her abilities as a fighter. As a child, despite her uniquely prodigal bending talents, she was treated as a defenseless little girl, a little princess in the most misogynistic use of the phrase. When Zuko was given a dagger by Iroh, she was given a doll dressed in the latest Earth Kingdom fashions, something she casts aside with a scoff and eye roll that seem far too knowing of the bounds of sexism for her young age.
On the other side of things, Katara has to deal with similar misogyny. Most of what she deals with is unspoken, the roles she is meant to subscribe to, which inherently work against her ability to learn how to fight and bend. This is directly and explicitly enforced when they head to the Northern Water Tribe to learn waterbending from an actual master. Aang is warmly welcomed, but Katara is shooed away by Pakku, who patronizes her and tells her to go learn healing with the other women. She is only allowed to train with him after physically fighting him to prove her talent (and after he finds out that her grandmother is the woman he once proposed to). Despite the strength of their bending (both masters at fourteen, both the best benders of their elements), they both have to deal with a kind of sexism that directly questions their abilities as fighters, leaders, and powerful young girls.
Ultimately their traumas are the things that either make or destroy them; their stories show the opposing ways that such similar trauma can function, how it can mold two distinctly different girls. Kya and her sacrifice make Katara the person she is. Rather than kill the person who took her mother or use bloodbending to her advantage, Katara shows mercy and kindness, understanding the value of human life and of compassion because of what her mother did. Azula’s mother ends up being her downfall; visions of her mother in the mirror make her even more unhinged and unstable. Azula loses it, challenging Zuko to an Agni Kai without her usual sort of clarity because of how Ursa was banished and left her behind with Ozai and his cruelty. It makes the narrative of Azula even more tragic, to see what she could have become had she been shown love rather than manipulation and abuse. It also makes the narrative of Katara stronger, to see what she might have been had she let her anger and vengefulness lead the way. They are both beautifully compelling characters and their relationship as foils to one another is one of the most interesting character dynamics in Avatar: The Last Airbender.