The Orthodox Church ceased full intercommunion with the Roman Catholic Church because of the RCC's demand that we adopt their system of church government and their version of the creed - this was a violation of Holy Tradition (to demand this of us), and so we refused and mutual excommunications were upheld. The history is much, much more messy than that, but that is the essence of it.
The issue really comes down to this: is the Papacy meant to be the juridical head of the Church?
I stress "juridical" because both East and West agree that the Papacy's role includes that of strong leadership - that Rome had traditionally been the Christian see of highest rank and importance. We tend to see this as a sort of "senate majority leader" ; one who leads from within and alongside (not OVER) the others. Furthermore, breaking rank on a particular issue with the senate majority leader may be unwise, but it hardly gets one kicked out of the senate. Each senatory, essentially, still has their one vote. The RCC, at the time of the schism, was describing the Papacy as a sort of super-monarch - literal RULER of both Church and State. I would refer you to Dictatus Papae, by Gregory Hildebrand, as an extreme example of this kind of rhetoric. Though not dogmatic in the RCC today, Dictatus Papae was representative of the general attitude of the Popes in the 11th - 13th c. (sometimes called the "imperial" papacy because of its active attempts to oversee the political life of Europe, including directing military action). There seems to be in mind here an almost theocratic feudal system, with the Pope at the top and the Kings on the political side, the Bishops on the Ecclesiastical side. You can see this by the Pope's claimed right to both appoint and dethrone kings and emperors (as well as bishops) during this time period.
Needless to say, that was not the model we'd known in the East, and we were not about to aquiesce to it just on the Pope's say so.
Furthermore, this wasn't the model of the early chuch; historians have vindicated the East on this point. If you go back to the earliest sources there isn't even evidence of there BEING a single bishop in Rome until the late 2nd century (150 - 180 AD). It can hardly be APOSTOLIC tradition (something completely binding on the entire church and worth excommunicating half of Christendom over) for the bishop of Rome to be fully in charge of the Church and State if the first 130-odd years didn't even HAVE a bishop of Rome.
Furthermore, once there was a monarchial bishop, there is no evidence of seeing this bishop as a unique successor of Peter until the late 3rd century. There is no known claim to overt Papal authority until the late 4th century (when Rome's political status as a city had begun to decline - that's when it starts to claim more ecclesiastical authority). It wasn't until the 5th c. that there was a Pope who actively claimed (as a significant element of his theology) authority over the other hierarchs - Pope St. Leo the Great in the 450's. Even then, if you go and look at the history, Pope St. Leo never had real, active authority outside of Italy with the exception of a few areas of southern France and Macedonia (northern Greece).
In the late fourth century, 381 AD, we find the first example of council and Pope in conflict. The Second Ecumenical Council, which upheld the Trinity against neo-Arian heretics, also recognized the growth of Constantinople in importance within the empire and granted it "Second Rank" but with "Equal Rights" to Rome (as a court of ecclesiastical appeal). Rome denied this, while accepting the rest of the council. It, however, became de-facto reality. Rome didn't recognize the canon until it had sacked Constantinople in 1204 AD (900 years later) and placed its own docile, pupet Patriarch on the ecclesiastical throne there. In that intermin 900 years Constantinople was the 2nd see of Christendom - 2nd to Rome in rank, but equal in rights. The council had operated against the Pope, saw nothing wrong with this, and the bulk of the church agreed.
In the fifth century, at Ephesus, the decisions of a local Roman council against Nestorius were not upheld as "de-facto" correct by virtue of Rome's opinion. Rather, the 3rd ecumenical council considered the matter for itself. It considered itself to have sufficient authority to overturn Rome's (and Alexandria's) decision if it chose to. The same happened in the mid-fifth century at the Fourth Ecumenical Council of Chalcedon, where, despite St. Leo's papal-legates demanding an immediate acceptance of his "Tome" on the meaning of the Incarnation, the council deliberated over the text with the full assumption of its right to disagree and overturn St. Leo's views. In the end, both councils did accept the same decision as the Pope (just like at the Council of Jerusalem in Acts 15), but their very method reveals their mind.
Things didn't turn out as rosy at the 5th and 6th ecumenical councils. In both, the Pope was directly censured and forced to relent. In the 5th, a still living Pope Vigilus (who directly argued against even calling the council) was declared a heretic by the council for refusing to condemn key Nestorian writings, and relented after being placed under house-arrest in Constantinople (not the most polite of councils...). The 6th ecumenical council post-humously declared Pope Honorius a heretic for actively teaching (if not inventing) the monothelite heresy (the doctrine of one "will" - a teaching of Honorius that built on an attempt to reunite with the Coptic church by describing Christ as possessed of a single "energy"; Honorius was the first to suggest expanding this to mean a single "will").
In short, the Popes were clearly (functionally) under the ultimate authority of an ecumenical council. Apart from such extreme examples, the church de-facto goverened itself in a decentralized way during this first millenium. There just wasn't any sense of papal infallibility (I challenge one to find even an inlking of this before the 7th c. AD), or real / functional papal centralism of the type the West tried to force down the East's throat in 1054 AD.
In the 9th and 10th c. (800 - 900) the Frankish and Germanic empires in Northern Europe hijacked the papacy and episcopate in the West through lay investiture and simony. During this time countless forgeries were circulated defending a central role for the papacy in church and state. Most notable were the Donation of Constantine and the Decretals of Pseudo-Isidore. These documents became the foundations of Western revisions of canon law during the Clunaic monastic revival of the 900's and the later Papal Reform movement of the 1000's AD.
In other words, the Papacy's actions in the 11th century, at the time of the schism, were unprecedented in scope and based on fake history. They were not Holy Tradition, and they have not been repented of to this day - rather they have been compounded by doctrines like Papal Infallibility and the militant Papacy of the Crusades and the 1400's. We see no reason to support such a thing, especially given the incredible division it has caused in the West (the Reformation and all that).
Incidently, during the 1400's, it was a Church council that solved (by the exercise of its own self-claimed authority) the Great Papal Schism. This council deposed Popes and antiPopes and selected a replacement single Pope. Even in the West, when push came to shove, the Councils still could throw their weight around.
Now, we don't believe that councils have any de-facto authority. There are plenty of fake councils. But we also certainly don't hang our entire hat on ONE see being authentically orthodox the way the RCC does on the Papacy.
What would you do if the Pope was teaching heresy? Where would the Church be if the Pope began to excommunicate people who didn't agree with his heresy? This is what the East feels happened to it - at that point, the Pope ceased to be orthodox; he fell into a heresy he remains in to this day. We lament it, but we will not shy away from its implications nor bow to it in the name of ecumenical good feelings. Please forgive us if that offends - we cannot do otherwise in good faith.
In Christ,
Macarius