Granted a lot of my knowledge of Ῥωμανία is from it's earlier portion (636-1081 or so) but I think for a model of the Roman state continuing into the modern era we need to look at the Russians. The Romans would suffer much of the same problems that Russia faced when trying to get "into Europe" so to speak. They would be viewed as "oriental" and "eastern" in the eyes of the West and would have a difficult time, at least before the 18th century, in participating in wider European politics. The dynamic would be to look at them as foreigners and schismatics. I think that much of the enmity between the Romans and the "Franks" would still continue at least past 1453 into the 16th and 17th centuries. Much like the Ottomans, perhaps powers like France would use them as a counterbalance against the Holy Roman Empire and, depending how much the Romans expand, they could play a big role in Italian politics once again. As shown by the Byzantine recovery in the 9th and 10th centuries on to the apogee under Basileos II, the manner in which the Romans deal with the Turks and Timur and how they expand will play an important role in how the state develops. I think as the state expands, the pull away from Constantinople will cause resentment much like it did in the 10th century. Cosmopolitan Romans of the capital will contest power with the land-owning provincials. Perhaps this trend would wane as technology got better and communication and travel allowed a closer cooperation and so a larger range of Constantinoplian projection, but maybe not. If you want a more developed answer you might need to provide additional details to the timeline and how it diverges but the premise here is interesting. The Republican institutions in the "Πολιτεία τῶν Ῥωμαίων" and their reaction to the passage of time and the modern world are also fascinating. The rise of the printing press I think would involve provincials far more in the horse-tradings inside the capital. Instead of the cosmopolitans often making or breaking a ruler, the provincials would receive news and ideas faster and easier. Constantinople could no longer be an inside political club commanding the provinces. Instead, the significance of public opinion in the themes would matter more since more citizens would be in the loop as to the political goings on and far faster. The powers of the Basileus would probably stay continuous as is the tradition, the bureaucracy likely get bigger with way more people to take care of and more mouths to feed. The relations with the East would be more hard to predict. The situation would largely depend on which powers exist there as always but if push comes to shove, the Taurus is likely the consistent political border upon which any further major incursions would fail.