Azula and Kuvira are separated by over 70 years, but they are remembered as some of the Avatar's mightiest foes. Both of them are fierce teenaged girls and the only people to defeat an Avatar in combat. Who would win in a battle between these prodigies?
Azula is the daughter of Fire Lord Ozai and Fire Lady Ursa. At the tender age of 14, she orchestrated a coup and captured Ba Sing Se, ending the war in a decisive Fire Nation victory. That very same day, she killed Avatar Aang (only for him to be revived) and took on waterbending master, Katara. On the day of Sozin's Comet, despite losing her sanity, she was able to take down her brother, Zuko, and force Katara on the defensive. While the battle ultimately resulted in her defeat, Azula showed she was a very powerful bender and will not soon be forgotten.
As a child, Azula excelled as a firebending prodigy. She has great control over her breathing and firebending, to the point where she was able generate blue flames and fight on equal ground with her uncle, 'The Dragon of the West.' Azula had mastery over another form of firebending - lightning generation. Azula's lightning is capable of inflicting great damage, but may not be lethal. Her comet-enhanced lightning blast incapacitated Zuko, but did not kill him.
She was able to fight with such great precision, a well-placed flame could end a battle. All flames were well-placed.
Azula can use her firebending for propulsion, whether it be for a faster charge, or even propelling herself like a rocket.
Azula wears strong metallic armour. Her armour is capable of stopping fire blasts, but has little use against blunt trauma impacts. That is, a powerful boulder would inflict great damage, and maybe even destroy the armour.
Azula's armour is light, allowing her to maintain her agility.
Aside from Azula's powerful bending, she has many other advantages. She is extremely agile, being able to evade Aang and Toph's attacks without her bending. She is also a cunning strategist, formulating the plan to defeat the Water Tribe invasion and conquer Ba Sing Se from the inside.
Princess Azula does not take defeat well. After she lost the battle of the Boiling Rock and her friends left her, she descended into insanity. When she begins to lose a battle, she will rapidly lose her sanity and with it, all her control and precision. Zuko and Katara were able to use this to their advantage on the day of Sozin's Comet and were able to best her in an Agni Kai.
Azula is also somewhat reliant on allies, being unable to fight alone. To conquer Ba Sing Se, she needed the help of her friends, Mai and Ty Lee. And taking down the Avatar was only possible with Zuko's intervention.
Kuvira was abandoned by her parents at age 8. She was adopted by Suyin Beifong, the daughter of the legendary Toph. Being a natural leader, she quickly rose through the ranks to become captain of Zaofu's security forces. After the death of the Earth Queen and the Earth Kingdom's descent into anarchy, Kuvira came to her nation's rescue and brought order to the divided nation. For her actions, she was awarded the title "The Great Uniter". In the battle of Zaofu, she was able to defeat Avatar Korra and take the city, thus uniting the kingdom. However, her greed got the better of her, and she was defeated when attacking Republic City.
Kuvira single-handedly defeated a horde of bandits.
Being trained by Suyin, Kuvira was a master earthbender. When the Red Lotus raided Zaofu, she fought on equal ground with Zaheer, the master airbender. The battle ended with Zaheer being forced on the defensive with the rest of his men. Kuvira's earthbending was also able to subdue Korra in a later battle.
Kuvira uses metalbending to form whips.
Not much is known about Kuvira's earthbending, because she rarely uses it. Kuvira is a master metalbender and makes extensive use of it. When Kuvira served in Suyin's guard, she made use of metal cables for whipping and subduing her enemies. By the time she took over the Earth Kingdom, she was able to subdue foes, simply by binding their hands and feet with metal strips. She could then use her metalbending to throw the strips, along with her foe. With her metalbending, she was able to levitate Varrick, who was wearing nothing but a metal collar.
At close range, Kuvira can use her metalbending to form punch-daggers from strips of metal on her sleeves.
Kuvira does not wear armour, save for the metal on her sleeves and collar. The metal can be bended into armour, but Kuvira favours agility over armour.
Kuvira trained her agility as part of Suyin's dance troupe.
Kuvira is very agile, being able to maintain balance while swingning on metal cables. She can evade attacks with minimal effort and return with her own attacks soon after. With her agility, she was able to defeat her teacher, Suyin, in battle.
Kuvira is a brilliant strategist and leader, being able to restore order in Ba Sing Se and reunify the entire Earth Kingdom (and maybe even United Republic, if not for Korra's intervention) in only three years.
Kuvira's weakness is her uncompromising desire to succeed. Kuvira went to great lengths to achieve her goal of unification, even going so far as to attack her fiancé and betray her adoptive family.
However, in battle, she does not appear to have any weaknesses and was arguably never defeated in battle.
Azula strode through the Earth King's throne room. She stopped, noticing Kuvira standing on the other end of the room. Azula had defeated Ba Sing Se's greatest earthbenders, and now felt compelled to face the last one. She landed the first blow, attacking Kuvira with a blue flame. Kuvira stepped to the side, avoiding it. Kuvira raised to stones from the ground and punched each one at Azula. Azula span to the sides, avoiding each blow. Azula kicked a stream of blue fire, narrowly missing Kuvira. As Azula placed her foot on the ground, Kuvira stomped, creating a trail of broken earth. Unwittingly, Azula stepped on the trail and fell to the ground. Kuvira trapped her in a pile of earth, trapping her, but exposing the head.
The earth pile exploded, Azula having escaped with bending. Using her firebending, Azula propelled herself to Kuvira. Kuvira launched herself into the air, using two columns. Azula stopped her charge and kicked, using all her fire to blast Kuvira out of the air. Kuvira landed with a thud. Azula stood over Kuvira, placing a foot on her chest and forming a flame in her hand. Kuvira raised her arm and clenched her fist. She stood up and jerked her arm up, levitating Azula by her metal armour. She began shrinking the metal, compressing Azula's ribs. While struggling, Azula managed to strike Kuvira with a fireball, thus knocking her over and freeing herself. In a hurry, Azula removed her armour.
Azula stood back and moved her arms in a circular motion. When she extended her arms, lightning crackled from her two fingers. Kuvira managed to avoid the first strike. Azula tried once more, this time, striking Kuvira in the metal on her arm. Kuvira acted by bending away the metal plate before the lightning could do its work. Kuvira's arm was injured, but the fight continued. Azula jumped on the wall. She jumped again, avoiding a metal strip. She jumped again, avoiding a rock. She jumped again, but Kuvira's cable coiled around her leg.
Azula, crashing to the ground, began to lose her sanity. She unleashed a flurry of powerful flames. Each one, however, was blocked by a rock pillar. Kuvira threw metal strips at Azula and bound her hands and feet together. She threw Azula into the wall, unwittingly loosening the strips. Kuvira collapsed the wall on Azula and walked to exit the room. Azula dug her way out of the pile and propelled herself once more. Kuvira kept walking, back to Azula. Azula came ever closer, lightning crackling in her arms. Kuvira continued to walk, paying no attention to Azula. When Azula came within striking distance, Kuvira turned, punching Azula's stomach with a dagger formed from her sleeve metal.
Azula's agility would often give her a significant advantage over a typical, stationary earthbender. However, Kuvira has been shown to have equal agility and can make use of quick attacks, particularly with metal.
One would be tempted the think that Azula's use of lightning would put a quick end to the battle. However, this is unlikely, as Azula does not utilise it to such a great extent and Kuvira does not wear enough metal. On the other hand, Kuvira's mastery over metalbending and ability to levitate a person with it gives Kuvira a great advantage and the ability to disarm Azula before she can even use lightning. Azula's armour is metallic, giving Kuvira an advantage.
In the end, however, it came down to jing. While Azula's control is great, her use of positive jing means she is constantly on the offensive, whereas Kuvira's use of negative jing means she can easily shift between defending and attack when the moment comes.