Originally posted by joak
The Byzantines had at least one serious problem, in that they really were considered "alien" by the rest of Europe, which meant they had no natural alliances with either other Christian or Moslem nations. This mistrust predates 1204--it probably made 1204 possible. Even at the very end, when you'd think they'd perceive a "common enemy," there was deep and often bitter distrust on both sides.
Because of that, I think there was a good chance the empire was going to be squeezed from both sides regardless, and it was going to be hard to recover if it faltered. If it did make it to modern times, it might well have filled almost the same roll as Turkey did--viewed as only partly a European nation, not quite part of the same system of dynastic and religious events that were shaping the modern west.
Despite genral mistrust the nation could probably gain a lot of prestinge one the Renaissance started (if it started of course, as the fall of Constantinople created a large influx of Greek knowledge into Italy). We may not forget that as the centuries passed by that there was an increasing ove for all things classical. It even came so far that there was a real Romano/Hellonophilia going on in the 17-19th centuries. And as the Byzantine Empire would have been recognised by the west as the remants of the Roman Empire (after a lot of debating and searching old record and stuff)... Well, they might have actually gotten some new support, maybe even an apology for 1204 (if that happened in the scenario).
Of course the Byzantine Empire had to survive first.
If it did adn maybe even crossed back into Asia Minor it could potentially have played a unifying role between Christians and Muslims in the area. The empire would most definitly have turned itself into an industrialised state (around Constantinople there would be large port, trade center and accompagnying industries), there would be (possibly) a great understanding with the Russian state (in whatever form, but no tzar) on religious grounds. This understanding could have prevented some or all Balkan conflicts, maybe even WW1.
I say maybe because nationalism would be a large factor in the Empire's stability (imagine all those Turks wanting their own state, or wanting to join their independent brethren in an independen, but smaller Ottoman Empire). There would be socialist movements, liberal movements, democratic movements (these would have their basis in the American and French revolutions, but also in the Athenian Expirement). All these things could very well trigger a world war, but with a different face this time. We could even see the fall of the Empire after such a war (as happened in Austria, Germany, Russia) and a change to a Republic Byzantine State, maybe even a loss of some territories, depending on what side they took in such a conflict.
Also, the name of the state would most definitly not be Byzantium, as they never called themselves that. What it would be i don't know, maybe Imperium Romaion (or something in such a fashion)?