Like Jimbrock said we don't know that much about what happened to begin with, but there's a decent chance not much would change. The Trojan War was essentially a loss for the Greeks in that they had lost so many men and basically left their country without any defenses for the 10 years. The only difference there would be the Troy surviving.
At some point, barring weird chaos-theory butterflies Tory would get nailed by the Persians eventually like every other power in Anatolia. And with a little luck the Persian-Greek wars would probably have turned out much the same way. Maybe the wars start start later because there are fewer Greek Anatolian colonies, since that would have been Tory's sphere of influence, which means that Athens doesn't help the Greek colonies which doesn't incur Persia's wrath, but eventually the Persians will come for Greece anyway.
Mythology-wise the Greeks have enough material to simply flesh out the era before the Trojan War (the era of Oedipus and the Seven Against Thebes) into any equally epic tale.
Finally in regards to Rome, there's no conclusive evidence that Aeneas actually existed to found Latium. However considering the penchant the Greeks and Phoenicians had for founding colonies in the Western Mediterranean, it's quite possible the Trojans would have founded an analogous colony anyway.