Different paths of historical development, basically.

The IRL Medieval Catholic Church benefited from a number of factors:

• It was the only pan-Western European (pan-European, when Rome and Constantinople could agree that I + I = δύο) institution when the Roman Empire fell and the first medieval kingdoms of the Franks, the Lombards, etc. were forming. Not only did that give it a certain amount of prestige, but it also meant that it was the only institution that could coordinate across borders, the only common authority that feuding kingdoms might appeal to.
• It was the largest landowner in Europe at a time when land was the major source of political, economic, social, and military power. And because it was a corporate landowner, unlike with feudal lords, land wasn’t given away as dowries or split between sons or sold off to pay for ransoms, and there were no cases of the land falling into escheat because the only heir died intestate. The corporate entity kept growing and growing, century after century, and so the estates consolidated and could take advantage of economies of scale and do really long-range investments, making the Church a real economic engine of the Middle Ages.
• It was also almost exclusively the source of literacy, learning, and communication. Churchmen were the literate class, especially early on, so in every court in Europe there were clergy serving as officials of state, bureaucrats, scribes and secretaries, as well as their religious duties. Since Church Latin was the only common tongue in Europe – the lingua franca well before diplomats started speaking French – the Church was immensely important in international communication. Up until the invention of the printing press, monks copying out manuscripts was basically the only source of books. 

The Faith had none of these advantages.

When the first fair-haired Andal pirates landed on the shores of Westeros, they brought the Faith of the Seven with them, but no institutions – there was no High Septon and no council of the Most Devout to exercise managerial control, no network of septs and septries dotted across the continent for the Faith to draw revenue and manpower from. Instead, the warlords and adventurers very much followed their own truth, carving the seven-pointed star on their chests and letting the Seven speak to them (and surprisingly, the Seven told them to go forth and carve themselves kingdoms). While holy men no doubt would have influence on religious matters, as long as the Andals were smiting the heathen and cutting down their weirwoods, there’s not a lot they could say to shape the actions of the warrior caste they were dependent on. 

For a brief period, the Faith could exercise some influence through the Arryn Kings’ patronage, but once the tide of Andals spilled out into the Riverlands, the petty kings and warlords and adventurers had no reason to listen to the King of the Mountains and Vale. And so the Faith would have to follow in their wake, building as they went. 

And then the Andal tide broke, first on the rocks of Moat Cailin and the equally stony shores of the North, and then again on the stable, powerful, and dynamic kingdoms of the West. Here were these foreign power structures, thousands of years old, who were assimilating into the Faith to be sure, but on their own terms and following their own interests, rather than the Faith’s terms and the Faith’s interests. So the Kings of the Rock and the Reach would become patrons of the Faith, but there would be no “Donation” of “Constantine”, no independent state. 

Moreover, the Faith would also have to deal with competition from another pan-continental corporate institution, one which had been operating for thousands of years, which controlled access to literacy, learning, and communication, and which had advisors whispering into the ears of every lord in Westeros: the Citadel of Maesters. Now, I believe that there was a compromise between the Faith and the Citadel (incidentally, if someone could send me an ask to remind me to explain why I think the Andal language was key to this compromise…), but it was one where the Citadel’s monopolies and jurisdictions would be respected. Septons and septas could teach basic literacy and the tenets of the Faith, but the rest would be the domain of the maesters.

So when I talked about the Dictatus Papae and the Walk to Canossa in previous asks, it’s actually a good example of how the two institutions were different. Here was Gregory VII, one of the most important Popes in history, laying down the law to cement the authority of the Catholic Church vis-a-vis the Holy Roman Emperor:

1. The Church is autonomous. Bishoprics and other offices belong to the Church alone, even if these positions had become mighty feudal states, Imperial Electors even. Only the Pope had jurisdiction over Church officials and lands, legal disputes involving the Church had to be settled by him in Rome, and so on.
2. The Pope is supreme over secular officials. Here Gregory really ran wild, stating that “all princes shall kiss the feet of the Pope alone,” and “it may be permitted to him to depose emperors.“ This, when Emperors had previously appointed and deposed Popes.
3. The Pope can dissolve the bonds of feudalism itself, through proclamations of excommunication, interdictionn, anethema, and so on. As Gregory put it, “He may absolve subjects from their fealty to wicked men.” And Gregory would do so, punishing Emperor Henry IV for attempting to assert authority over the Prince-Bishoprics of the Holy Roman Empire. 
4. “The Roman Church has never erred. Nor will it err, to all eternity–Scripture being witness.“  That’s the origin of Papal infallibility, although less immediately relevant to the crisis.

This was somewhat controversial, to say the least. Henry IV responded by declaring Gregory “at present not pope but false monk,” calling for a new election of the Papacy, and challenging Gregory: "I, Henry, king by the grace of God, with all of my Bishops, say to you, come down, come down!” And while the Papally-sponsored rebellions in 1073-1075, and then again from 1077-1088 did force Henry IV to do penance in the snow at Canossa, in 1080 Henry IV was ready to fight back.

In 1080, Henry IV proclaimed Clement III to be the true Pope, reasserting Imperial authority to name the Pope, after Gregory had blessed Rudolf von Rheinfeld, the elected Emperor of the rebels. The next year, Henry invaded Rome and when Gregory VII called upon his Norman allies from southern Italy, they promptly sacked the city and Gregory was forced to flee when the citizens of Rome rose up against him, and died in exile. 

If that’s what happened to a Catholic Church that was far more powerful than the Faith ever was, imagine what would have happened to a High Septon who tried to pull a stunt like that against the Lord of Oldtown and the King of the Reach. It wouldn’t have been the first time a High Septon was assassinated, and it wouldn’t be the last.