One of the most important characteristics of Christianity (and Islam) is that spreading the religion is part of the doctorine itself. This is why Christianity grew from a little religion from a backwater town in the eastern part of the Roman Empire to the dominant religion in Rome in only a few centuries. The Hellenistic polytheistic religion didn't supplant well; much of the Roman Empire outside of Italy, Greece, and parts of Turkey didn't follow Hellenism. Without Christianity, we would be seeing a Europe divided religiously. With the fall of the Roman Empire, Hellenistic polytheism would likely lose traction in Italy, though it might not actually die out. If anything, Germanic people moving into the area may fuse their deities with Roman deities, creating a sort of hybrid polytheism.
Returning to my point of Christianity spreading itself, a religiously diverse Europe would certainly complicate things. While there would certainly be religious differences, it would not be to the extent one might think. Most polytheistic religions don't emphasize on spreading the doctorine. Germanic tribes invaded Rome because it was seen as a better place to live than in Germany rather than religious differences. Norsemen invaded Europe and Britain because they were more arable and the cities contained vast amounts of wealth. Without Christianity's spread, we would see the evolution of polytheistic religions. Some would likely dwindle (British Polytheism comes to mind) while others would fuse (again, Hellenism and Germanic polytheism. Let's call that Lombardic Polytheism). To sum it up, religious feuds, while possible, are less likely than one would assume. European politcs would likely be a lot more heated without nations finding a common ground, though.
I don't see Zoroastrianism spreading nearly as much as Islam. It would likely spread throughout the Persian area to be the dominant religion there, but it wouldn't be in Northern Africa at all. Hinduism in India would likely reach into the northern reaches of the nation, meaning less Hindu-Muslim violence, though there may be some cultural clashes between Hinduism and Zoroastrianism.
Sadly, without one religion bringing people together, religious golden ages such as the Muslim Golden Age would not have happened. Say what you will about organized religion, but it has caused quite a bit of good in the world. Though some people believe religion has held society back, I believe that it helped jumpstart western intellectualism rather than retard it. Without Christianity or Islam and the unity that came with it, we would have progressed at a much slower rate overall in the scientific realm. Furthermore, many people came together to fight the spread of one spreading religion. Europe would basically be a cluster of nations with no reason to work together at all.
Persia would likely see a bit of progress since they'd likely be unified under Zoroastrianism, but I doubt they would see such a profound golden age seen by Islam during the Muslim Golden Age.
Eastern Asia would be a massive center of power. During the Middle Ages, Tang China was expanding west towards the Abbasid Caliphate. Their expansion was halted at the Battle of the Talas River. Without the Abbasid Caliphate, China would likely expand farther. How far would they get? Who knows? How long would they be able to hold onto this territory? Who knows? What is likely is that China would be a lot more powerful (even moreso than it already was).
tl;dr Europe would be meh, Middle East would be meh, China would be living it up