I'll try my first attempt at a "historian" quality answer since this is in my field of interest and nobody else answered for a week.

So geographic areas are split into Holy Sees, and starting in 325 with the council of Nicaea, the See of Rome is regarded as the superior See, and thus the Bishop of Rome is the top bishop, and considered God's representative on earth - other bishops were just men, but the Pope was basically as close to divine as one could be in Christianity. That council says Alexandria and Antioch are both tied for second. By 381 the council of Constantinople raises Constantinople to #2 though Alexandria complains. By 451 the council of Chalcedon confirms Constantinople as #2 after Rome and gives it jurisdiction over Alexandria and Antioch and also all barbaric territory and anything else that isn't in Rome's western territory.

The Chalcedon Council caused a problem with some of its theological points which lead to Alexandria breaking off and becoming what would eventually be recognized as Coptic. Other areas referred to as the "oriental orthodoxy" also broke away (Armenia, Syria, and Ethiopia).

So right there, the Pope has religious jurisdiction so to speak over all western Europe, while the Eastern church was a #2 and losing ground.

Nevertheless, the western empire falls and the Pope has to repeatedly call on the eastern empire to rescue Rome. Move ahead to 771, the Pope is threatened by the Lombards, Eastern empire won't help because the empress Irene isn't a fan of the Pope. And do the Pope asks Charlemagne for help. Charlemagne was at that time ruling a kingdom from France into modern Germany. Charlemagne helps the Pope. Over the next 30 years instability in the region continues. So in 800 there's a significant event. The Pope crowns Charlemagne as emperor of the Roman Empire.

The brilliant move brought legitimacy to Charlemagne's rule, and was a power play by the Pope - because he took it upon himself to presume he had the power to do it. When Charlemagne dies, his sons split the empire in three pieces (the middle one is eventually split and absorbed by France and the Holy Roman Empire). But it meant that the right to rule France and the HRE derived from the Pope's coronation of Charlemagne. Throughout Western European history, monarchs were crowned by bishops, representing the divine right to rule. Rome meanwhile maintained a degree of independence from the greater powers of Western Europe - it's harder to dismiss a Pope when you'd have to invade and you'd piss off the other powers. Not that it didn't happen - but the balance of power in Western Europe relied somewhat on the Pope.

Now look at the Eastern empire. By 800 it didn't need legitimacy, its legitimacy came from the fact that it was the Roman Empire and had been ruling over 1000 years. Meanwhile it lost much of its territory in the middle east, and was limited to essentially the Balkans and Asia Minor. Orthodoxy spread north into Russia, but the Eastern empire did not. The patriarch was within the Empire, and so the religion and government were more closely connected. The Eastern emperor could replace a patriarch without upsetting the balance of power in the region.

By 1200 the Empire was falling, and completely ended in 1453 when the Ottomans took over. The Ottomans were Muslim but somewhat tolerant of the Christians. They too wanted the legitimacy of the Roman Empire - the Ottoman viewed themselves as the inheritors of Rome and that they just shifted the religious beliefs of the Empire. But in order to do that, they kept the patriarch of Constantinople around. The patriarch sanctioned Mehmed as rightful successor of the throne, and Mehmed let the Christians live. The patriarch became a deputy of sorts to the Sultan and allowed to rule over the Christians in the Ottoman empire.

The Ottomans conquered the old ecclesiastical areas of Antioch, Jerusalem, and Alexandria, and if you recall, the patriarch of Constantinople is in charge of all of them. So basically the Patriarch of Constantinople wielded great power over the Christians in the Ottoman empire. For awhile the patriarch had some considerable influence on Russian orthodoxy, but that church split off in 1448, it still considered itself part of the Orthodoxy, but Moscow was its own ecclesiastical territory, independent from Constantinople.

But until colonialism spread Roman catholicism far and wide, there really weren't any other Christians in power around besides the western Roman Catholics. So no ability for the patriarch to expand his influence.

Sources - https://www.amazon.com/Byzantine-Empire-Browning/dp/0813207541