The Fire Nation without the avatar's return.
If it weren't for Katara's outburst revealing the iceberg, Zuko would have continued scouring the world, looking for the avatar. He would've investigated every unnatural light on his path, would've visited and revisited the air temples millions of times. He would've hung around the water tribes, hoping that the avatar cycle would come to its next step, he would've trekked the Earth Kingdom with the same hopes.
Maybe he would've become exhausted of his search sooner, maybe he would've let Iroh's wisdom get to him earlier, letting him start a humble life away from the Fire Nation...
No, that's unlikely. With Sozin's comet approaching so soon, Zuko simply wouldn't have the time to let go of the love he harbors for his country, unless stripped of all the tiny conveniences he had on his ship (something that likely wouldn't happen without the avatar's appearance setting the plot in motion). By the next summer, Zuko would still wander, all for the sake of getting his father's acceptance back, of getting his right to the throne back.
It's unclear if Azula would enlist her banished brother's help, especially since he would be so insignificant in the grand scheme of things without an avatar to capture. In fact, without Iroh and Zuko becoming traitors, it's unclear if Azula herself would leave her father's side.
Would she conquer Ba Sing Se without the Gaang giving her the tools to do so? After all, before their appearance in the city, there wasn't a clear path to invading it, since the Kyoshi warriors weren't deemed friendly yet (and without the Gaang visiting Kyoshi Island earlier, the Kyoshi warriors wouldn't have been near the impenetrable city either). Though perhaps she'd be tasked with taking the city during the comet.
There wouldn't be an invasion during the Eclipse without Sokka seeking out information about it and charting a plan. It would've been just 8 minutes of mild inconvenience for the Fire Nation. Only Bumi might affect something by taking over his city, but without Aang, the Fire Nation would regain it later anyway.
During the comet, Ozai's incineration force would be the same. Though Azula would greet Ba Sing Se instead of the White Lotus. Together they'd achieve world domination.
Ozai would be crowned Phoenix King. Azula would be his Fire Lord.
Depending on his luck, Prince Zuko might become a casualty of war, incinerated by his own people. His banishment might instead extend to the entirety of the planet, as every land would become Fire Nation soil (except, maybe, the Northern pole). Or the Phoenix King would let his son chase around that mythical avatar, just for the heck of it. It's not like an avatar would be able to reverse all this damage, but it's not like he wants his elder son back either.
In the best case, of course, there's the royal family pardoning Zuko, yet not restoring his titles. But I find the chance slim.
Though with Iroh being a good person, there's the scenario of him committing treason to thwart his brother's plans, once he learns of them. That would put both him and his nephew in a hot spot, increasing the chances of their banishment or death sentences.
To sum all of this up, a world without Aang's return would be really, really miserable and that's just covering the impact he had surrounding the Fire Nation's military conquests and the Fire Nation's royalty. Without the avatar setting on his belated journey and bringing his gang together, by chance or circumstance, the world would remain stagnant, suffering from its endless war. From a writing standpoint, it is great that the characters had such a lasting impact on their world! That's how stories should work!
It's still depressing to think what it would be like if they didn't set on their adventure though.
