One other aspect of keeping the Empire around is how you want to deal with the issue of slavery and small land holders.
The Romans in the west had massive problems because of their reliance on slaves. Even when it worked well it still meant they had no reason to develop power source technologies. If they did get rid of slaves, Christians aren't pro-slavery generally, and kept their cities from depopulating we might see a much higher level of technology.
In both West and East, the empire had major problems balancing the concerns of the small land holders and artisans who made up the armies and the majority of the economy with the large land holders and merchants who controlled the government and most of the trade-routes. Their tilt away from the interests of the small holders made it very hard to maintain the dynamic fighting forces and colonization models they had used so successfully in their early years.
If you could get them to drift the other way you would see a Europe of a very different color. Including super strong nations and armies. Like China only with powerful guilds and peasants who were a lot less starving and impotent.
The church would be even more dominated by politics than you would normally see in the west and probably have a lot less high level hierarchy. Though local bishops would likely have a lot of power. Religion would likely be a lot more mystical and challenged by philosophy rather than complemented by it.
I would predict the emergence of civilized German kingdoms which would either be assimilated or turn into satellite nations or antagonists.
A strong empire would change two major migrations at least:
Vikings- into Russia in a much larger way, probably come to dominate Scotland and might make more efforts to colonize Vinland.
Arabs- would see greater emphasis on moving into sub-saharan Africa since its better than facing Byzantine/Persian pressure.
Mongols would last longer with Rome as a trading partner and another pressure on the Arabs, Europe was never something they were very interested in, and would probably become Christian, they nearly converted anyways and with a strong ally to gain...