Certainly the popular image (a relic of Protestant historiography) of the Pope having an iron control over Europe and constantly being able to boss kings and emperors around is inaccurate. At the same time, while the Pope's lack of his own financial and military muscle and rather shaky hold over the largely autonomous barons and urban republics of the Papal States until the 14th century, when the Papacy acquired permanent professional armies of condottieri (mercenaries) and the permanent backing of Florentine bankers, made him quite vulnerable, that doesn't mean that in terms of his spiritual powers and personal charisma he couldn't be a force to be reckoned with. Let's take a look at two examples from the High Middle Ages, the period normally seen as the high watermark of the medieval papacy.

Let's start with the example of Pope Gregory VII (reigned 1073 - 1085). Gregory VII is probably one of the most famous medieval popes, above all for making the German king (and claimant to the title of Holy Roman Emperor) Henry IV kneel before him in the snow at Canossa (a castle in the Apennine mountains in Italy) for five days before finally granting him forgiveness.

Here's some context. Both the pope and the German king were interested in reforming the church but they had different visions for it. Henry IV believed that church reform should be a royal-led project as it had been under his predecessors going back to Charlemagne and Otto I, whilst Pope Gregory VII, who was at the head of a papacy which for the last 30 years had been becoming increasingly activist, saw it has his prerogative. Both claimed the right to appoint bishops, which until then had been the prerogative of kings and emperors. Henry IV believed that was his right because it was clear from the Bible (i.e. "Render unto Caesar what is Caesar's, Render unto God what is God's", "Fear God, Honour the King" etc) and Church history (i.e. the miraculous conversion of the Roman Emperor Constantine the Great in 312) that royal authority was sacred and directly came from God. On a more practical level, Henry needed to be able to appoint bishops because they were the closest thing he had to professional administrators, being highly educated men versed in Cicero's De Officiis and other ancient Roman models of statesmanship that were entrusted with cities, territories, castles and mints to govern in the king's name, and so it was essential that these men were as loyal as possible by literally owing their position to him. Gregory, on the other hand, believed that the New Testament showed that the Church should be free i.e. that Christ hadn't bled on the cross so that emperors could appoint bishops and boss them around, and identified lay investiture (the appointment of bishops and other clerics by kings and other secular authorities) as the main cause of corruption in the church. From 1075, they both also wanted to specifically control the appointment of the bishop of Milan, where there was the risk of the reform movement going out of control i.e. popular mobs known as the Patarenes were trying to lynch priests who were married (an accepted practice up until the 11th century i.e. St Paul believed that a bishop should simply be a sober man with one wife)/ had allegedly bought their offices (the sin of simony).

What ended up happening was that Henry threatened Gregory with deposition (all of Henry's predecessors since Otto I had deposed a pope at some point), calling him "the false monk Hildebrand (Gregory's birth name)", whilst Gregory threatened to denounce Henry as a tyrant and exhort the German dukes to rebel against him. Henry was persuaded by his wife and mother that he should do penance before the pope as good diplomatic and PR move (tyrants don't tend to admit they've been wrong). And so in 1077 the penance at Canossa happened and pope and king were reconciled. But after that Gregory reneged on his word, and the German nobility rebelled with his encouragement, electing Rudolf of Rheinfelden as their anti-king. Henry defeated and killed Rudolf in battle in 1080, and in 1082 he besieged Rome, captured Gregory VII and elected his own anti-pope, Clement III. Gregory VII was rescued by the Normans led by Robert Guiscard in 1084, but he died in exile. So from Henry IV and Gregory VII's story, we can see how it could be both smart and not smart to obey a pope - smart because of the pope's spiritual authority posing a potential threat to your own royal authority over your subjects, and not smart because the pope himself was very vulnerable and could easily be overcome with superior military force and individual popes like Gregory VII could be very untrustworthy and difficult to deal with by more peaceful, diplomatic means.

Let's take a different example, that of King Philip Augustus and Pope Innocent III (generally reckoned to have been the most powerful medieval popes). Philip had repudiated his wife, Ingeborg of Denmark, in 1193 after spending only a night with her, essentially because he couldn't stand sleeping with her due to her bad breath though Philip was also able to make up a spurious claim that their marriage was within the degrees of kinship forbidden by canon law and was thus incestuous, and began a bigamous relationship with Agnes of Meranie. Innocent's predecessor, Celestine III, disapproved but did virtually nothing. However, his successor Innocent III was a professional lawyer and a much tougher man. He declared that since Philip had not properly divorced Ingeborg, his marriage to Agnes was void and unless he repudiate her and take Ingeborg back there would be consequences. Philip refused and was excommunicated and the kingdom of France was placed under interdict (church services could not take place) from 1199 to 1200. However, since Philip was still friendly with King John of England and his Angevin Empire and the Holy Roman Empire was in a civil war, he didn't need to worry about any neighbours using this as a go ahead to invade France. Agnes died in 1201 and the excommunication was lifted , but Philip refused to take her back until 1213 when he needed the Pope's blessing to give his son, the future Louis VIII, a casus belli to invade England - King John himself had recently chosen to reconcile with the Pope after a four year excommunication. Here again we see that it was really a matter of individual circumstances whether monarchs chose to obey the pope or not - if it suited their interests it was the obvious thing to do, but if it didn't then so long as they lacked powerful and hostile rivals, within or without, they could basically get away with it though not without temporarily putting their subjects in spiritual jeopardy for a short period of time with the interdict.