What were some differences between the Popes and Holy Roman Emperors?
The primary disagreement between Popes and Holy Roman Emperors, which continued for a number of years between several Popes and Emperors was over division of power. Its origins went back to the crowning of Charlemagne as "Emperor of the Romans" on Christmas Day, 800 C.E. Did the Pope "make" Charlemagne Emperor? If so, the Emperor was subject to the Pope's authority, as he could "unmake" him. Or, did the Pope simply acknowledge that Charlemagne was Emperor of the Romans. If so, the Pope had no authority over him in his official capacity. Popes claimed that as the Vicar of Christ on Earth, their authority extended over all matters, both spiritual and temporal. It was perhaps best expressed in Pope Boniface VIII in his famous bull, Unam Sanctum:
Therefore, if the earthly power errs, it shall be judged by the spiritual power, if a spiritual power errs, it shall be judged by its superior; but if the supreme spiritual power errs, it can be judged only by God, not by man....Therefore we declare, state, define and pronounce that it is altogether necessary for salvation for every human creature to be subject to the Roman Pontiff.
Emperors, needless to say, resisted the supremacy of Papal authority. At times, the argument became somewhat petulant: Frederick Barbarossa refused to hold the reins of the Pope's horse as he believed it would indicate servitude on his part. Insults between the two were rife. Barbarossa, in responding to a Papal letter, commented:
We cannot wonder enough at finding your words insipid with swollen pride rather than seasoned with the salt of wisdom....You boastfully declare that by you I have been summoned; that by you I have been made first a citizen and then the prince; that from you I have received what was yours. How lacking in reason, how void of truth this novel utterance is, may be left to your own judgment and to the decision of men of wisdom!
The dispute ultimately devolved into other issues: Should the Emperor have the right to designate church officials, or at least veto the Pope's choices, or did that power belong to the Pope alone? This was the famous Investiture Controversy which raged for some time. Another issue: should the Emperor have the right to tax the church's considerable properties?
The dispute is far too detailed to be dealt with in this brief response; however it ultimately was concluded when Philip IV of France, (Philip le Bel, or "the fair.") actually imprisoned Pope Boniface VIII. Later, he compelled the Popes to take up residency at Avignon, the "Babylonian Captivity" of the Medieval period.
An excellent discussion of the controversy can be found in The Crisis of Church and State 1050-1300 by Brian Tierney.
This is a good question and the answer of this question really depends on what epoch of history that you are dealing with. However, generally speaking we can say that the pope and the emperor differed in two main ways.
First, the pope was the supreme leader of the Catholic church, whereas the emperor was only leader over the government. However, what complicates things a bit is that the emperors got involved in the issues of the church and the pope got into the business of the secular government. There was a lot of power play and at one point the pope was ever more powerful that the emperor. However, from an official point of view, they had jurisdiction over different areas.
Second, until the Reformation the pope's influence was very much international. His scope of influence transcended boundaries as Europe was Christian. However, when it came to secular leaders, they were limited to their own territories.