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Sozin's Comet honestly doesn't change that much; only two real differences.

The magic rock is gone. Instead, we get a flashback to Guru Pathik, and Aang realizes that he has to let go of his attachment to Katara.

The Maiko/Kataang victory laps in the last 5 minutes are gone. Mai and Zuko get a scene where they wish each other well, but acknowledge that they're over. Katara and Aang have a nice moment where they choose friendship. There's love there, but it's Philia, not Eros.

@burst-of-iridescent has a delightful essay series on Zutara, and the part that sticks in my head is that in the run-up to the finale, Aang and Katara are at their least unified. To wit:

In "The Southern Raiders" Aang is preachy, condescending, and more than a little hypocritical about Katara's plan to take vengeance for her mother. The episode ends with Katara explicitly rejecting Aang's belief that Yon Rha was forgiven.

In "The Ember Island Players" Aang is distraught at the idea that EIP!Katara's statements, namely that Aang is like her brother and she's attracted to Zuko, are true. When he questions Real!Katara about this, he responds to her statement that she's confused about her feelings by kissing her. The kiss is not returned. Again, no resolution is had.

Finally, in "Sozin's Comet Part 1" Katara is part of the chorus condemning Aang for refusing to even consider killing Ozai, no matter how many people are at stake. He runs off from the group, and from there disappears into the Spirit World to get Lion-Turtled. Yet again, no resolution, and the two don't reunite until the tea shop.

Now speaking of the Lion Turtles, I'm actually not opposed to them. Yes, they come out of nowhere to deliver an 11th Hour Superpower that handily spares Aang from having to actually make a choice he disagrees with, but at the end of the day it is a kid's show. Nickelodeon was never going to approve a script where Aang killed Ozai. Throw in a little bit of foreshadowing, and I'm good. It's worth noting here that the story of Avatar Wan was supposed to be covered in A:tLA, which would handily cover that requirement.

Now, for the post-canon. We'll start with Fire.

Zuko is NOT left alone in the Fire Nation. Similarly, Iroh does NOT fuck off back to Ba Sing Se.

Toph and Suki stick around. Suki in her canon role as commander of Zuko's Kyoshi Warrior bodyguard, while Toph and Mai use Toph's lie-detection and Mai's insider knowledge to purge threats to the new peace.

Toph eventually goes back to the Earth Kingdom to start a metalbending academy, but first she needs to make sure that her Sparky lives to be the grumpy old man he was born to be.

While Iroh is correct that for political and diplomatic reasons Zuko needs to be Fire Lord, he also bows to the reality that Zuko is plain and simply not ready to be the Fire Lord.

Zuko went from 4th in line to 1st in line basically overnight, and the 5-ish years he spent as Crown Prince were clearly not spent preparing him to succeed Ozai.

So a teenager with a fairly surface-level understanding of "how to monarch" has to self-Reconstruction the Fire Nation, while paying reparations, without having been militarily conquered.

This is how idealists get assassinated. New Plan!

Zuko is crowned Fire Lord. Iroh is his Prince Regent. It's very clear to all involved that Zuko is the one charting the course forward for the Fire Nation, while Iroh is there to convert intent to action, while teaching Zuko how it's done.

It doesn't hurt that Iroh is one of the Fire Nation's most successful military commanders, so the civil war route is a lot riskier for anyone to attempt.

Next, Sokka

Sokka honestly has a pretty good arc in the post-canon. Nothing I really feel the need to correct.

Eventually, Suki is able to hand off her duties in the Fire Nation to someone else and goes home to Kyoshi Island

It’s still home, but it isn’t the same. Or rather, she isn’t the same.

She never leaves the island behind, but it’s usually a stop on the journey from Wolf Cove to Republic City.

Then, Aang

Aang divides his time between Avatar duties and Last Airbender duties.

Avatar duties involve a lot of sitting in on meetings and reminding people that the ultimate goal is peace.

Last Airbender duties involve a lot of teaching Air Acolytes everything he remembers from his childhood. He gets lucky here, though.

The Airbender Genocide wasn't complete. More than a few Air Nomads escaped the Genocide, and hid themselves away. Some in small villages built in remote mountain valleys, others blended into Earth Kingdom settlements.

Plenty of quarter- or eighth-Air Nomad kids running around with airbending potential they never had the knowledge to develop. Think very early Book 1 Katara here.

The result is that a resurgent Air Nation is being formed, with a culture woven from the threads that survived through relics, the refugees, and Aang himself.

Airbenders are still rare, and it's over a decade before another airbender earns their mastery, but it's not his son and his grandkids when Korra comes around.

Finally, Katara

Katara spends a lot of time traveling. She spends time in the South Pole, helping to rebuild and learning Southern Style Waterbending from the released waterbenders. She also travels the Fire Nation and Earth Kingdom, doing what she can to solve problems.

There are a lot of problems to solve.

Her travels in the Fire Nation are particularly fruitful; word of the Last Agni Kai has spread, so she is known to be the one that the Fire Lord sacrificed himself for, and who healed his wound.

She and Zuko stay in contact, allowing themselves to have a slow-burn courtship.

After a couple of years she moves to Caldera City and starts getting down to seriously courting Zuko, preparing herself for Fire Ladydom.

The year before Iroh is set to retire as Regent, she and Zuko start thinking about the wedding.

There are a lot of potential traditions to uphold, even just between the Southern Water Tribe and the Fire Nation

This goes double for the daughter of the head chieftain of the Southern Water Tribe, and the Fire Lord himself.

They decide to have fun with it and do everything.

Aang presides over a private ceremony, family/close friends only, that is really just a mutual declaration of love and friendship.

Then come the Southern Water traditions. It's the full gamut, with ice-dodging, sacred hunts and more. In the end, Zuko is an honorary member of the Southern Water Tribe, and he and Katara are wed (again).

There's a diplomatic tour through the Earth Kingdom, stopping at Kyoshi, Gaoling, Omashu, Ba Sing Se, the Foggy Swamp the former Fire Colonies, and ending at the Northern Water Tribe. The language used artfully slides over whether the couple is newly married or about to be married, but overall it works well for the Fire Nation's reputation abroad.

The final act is in the Fire Nation. A full Royal Wedding, a grand affair of state, held at high noon on the day of the summer solstice. When all is said and done, Zuko and Katara now rule alone as Fire Lord and Fire Lady.