This sort of happened in OTL with the Neo-Hittite States like Carchemish lasting until about 700 BC. But, I'm assuming you mean a united Hittite Empire lasting for that time. A POD could be the Hittites win the battle of Nihiriya.
A united Empire would have been a bridge between the Greek and Middle Eastern worlds. It could have had a different effect on the Trojan War. But they were never able to conquer Egypt or Assyria before, so I doubt they could in another three centuries.
It also might have kept out Phrygian and Armenian migrations so the history of the Caucasus is different, possibly with an Urartian-speaking population surviving.