I've been thinking a bit recently about why fantasy novels are set in medieval worlds. Or even if they're not properly fantasy, if we want a story told about times before modernity, it always goes to the tales of knights and castles and kingdoms. I think there's fascinating ideas out there from even earlier, of the Roman Republic and the Athenian philosopher-kings, and I've written up a few points about what would make a story essentially classical instead of medieval: a necessary task if we're leaving the real timeline.
* Urban vs. rural: I think this is an element that makes things particularly compelling. The castle is replaced by the polis for affairs of state. I think the results of this is a much larger political universe: medievalist fantasy usually focuses on the conflicts between the king and the nobles, or the various squabbles of competing petty warlords. There's a lot more to deal with if you're in charge of a city - fully capable political classes to contest authority, and powerful nonpolitical actors: guilds, mobs, organized crime, artists, beggars, merchants. And of course, those things are all interesting to explore beyond political struggles - there are more people doing more different things in a city than in a pastoral setting, where you're either a knight or a farmer. Maybe an innkeeper.
* Power of the state: There's a bit of a paradox here - the classical ruler is more checked by rival powers in the city, but simultaneously much more potential authority, should he dominate those elements. Medieval kingships tended to be low-stakes affairs, and royalty was its own prize. The greater organization of a Roman-style state opens up a lot more avenues for characters to interact with the state, both successfully or otherwise. Or attempt to interact with the populace, should they be rulers themselves.
* Range and diversity: I think in a classical setting, your characters are much more able to travel. A medieval affair tends to concern itself in one realm, which is fairly homogenous. Rome encompassed Egypt and England alike, and you are able to regularly mix things up with different cultures, religions, or powers running across things. You have cosmopolitan characters able to seek these things out, too, and more places to flee. At the same time, there's much more of a continuum between insiders and outsiders. A foreigner is not nearly as obvious.
*Monarchy vs. republic: A medieval setting almost never considers the idea that maybe people shouldn't be ruled by a king. A classical setting can really take up this argument, all the more urgent if the choices are a republic or an empire. Warlords can aspire to more power by claiming the imperium, and dissidents can do more than support the just and fair younger brother of the king. You get to inject very potent, resonant idealisms into a conflict over power.
*Barbarians at the gates: Always a loaded question, I think in a classical setting you can explore more the line between civilization and barbarism - and at the same time, undermine that distinction. As above, it's not an insular society like so many medieval kingdoms, instead you have a very distinct line that separates the Romans from the Germans, which characters are going to lean on. It's also a way to empower emperor-types, who are going to win prestige by turning back the hordes, and use fear of same to command their authority. Their supporters will be more complex than the toadies of the evil king.
*Thinkers: more than plotters, you get to have characters who are philosophers, and able to influence events with ideas. With consequences as disastrous as anything else, of course, as Plato, Socrates, and Aristotle were able to witness.
*Atmospherics: Here's where we get the robes-for-chain mail swap, but some of this is interesting in its own right. You get to move away from the colder north lands to the lush Mediterranean, and from wood and stone to marble. Villas. You can play a lot with tone here.