During the period prior to the East-West schism, the Popes and the Patriarchs of Constantinople, were, in a sense, both colleagues and rivals.

The Holy See in Rome enjoyed a certain preeminence due to its claim to be associated with the Apostles Peter and Paul. The connection to Peter was especially important, as Roman bishops asserted the Christ handed over the "keys to the kingdom" to Peter and his successors (the Popes) in perpetuity. This claim to primacy was not unrecognized in Constantinople, but the eastern churches generally held a much weaker of view of what the primacy of the Roman bishop actually meant. Popes were said to enjoy a certain primacy of honor, and little beyond that.

All of this is complicated by the involvement of Roman emperors in this puzzle. After Constantine I relocated the imperial capital to Constantinople, and as the political center of gravity shifted, so did that of church authority. The Constantinopolitan patriarchs proximity to the imperial office played an important role in its rise to prominence. Rome and Constantinople shared the distinction with Antioch, Jerusalem, and Alexandria as being the empire's great pentarchal sees, and Constantinople's status as the imperial city gave its patriarch a certain gravitas in church affairs in the east. Rome, as the only pentarchal see in the west, was the most predominant church there.

After Justinian's reconquest of Italy, the position of the Pope in Rome became heavily dependent on the secular power of the eastern Roman Empire. Throughout this period, lasting into the middle of the eighth century, the Bishop of Rome was essentially nominated by the emperor, and later the Roman Exarch in Ravenna. As a result, Popes did not enjoy much in the way of preeminence over the Patriarch in Constantinople. Popes and Patriarchs competed for jurisdiction over churches in the central Mediterranean and along the Adriatic, and it is during this period that the Patriarchs in Constantinople started using the descriptor "Ecumenical," in way a kind of claim to universal jurisdiction.

Over the course of the eighth century, as Roman emperors dealt with troubles on their borders in the east, and as Lombard pressure resumed on Italy in the west that Rome and Constantinople drifted further and further apart. After Emperor Leo III seized papal lands in Italy and Sicily, and reassigned some church jurisdiction from the Pope to the eastern Patriarch, one of the critical links between the Pope and the Empire had been severed, and as the Lombards and later the Franks became more important, Popes had fewer reasons to concern themselves with opinions in Constantinople.

Over the intervening centuries, the Latin and Greek churches drifted further and further apart, as the Papacy established itself as a political entity on the Italian peninsula and as a player in western political affairs. As stated above, the east had its own problems to worry about. As knowledge of Greek - the dominant language in the eastern church - faded in the west, regular communication between Greek and Latin churches broke down, and the theological traditions interacted with each other far less. By the time the schism occurred in 1054, the churches had been drifting apart for centuries.

Readings

Geoffrey Barraclough, The Medieval Papacy

Timothy Ware, The Orthodox Church: An Introduction to Eastern Christianity. (This one is by a Greek Orthodox bishop, so he definitely has his own take on things, but it does contain some commentary on Orthodox views of the Papacy)

J.N.D. Kelly, Early Christian Doctrines