There are many historical factors that went into the making of the Eastern Schism or break from the Papacy, but the main differences were doctrinal.
1) With the Schism of 1054, the Greeks Church (Orthodox) assigned to the Pope a primacy of honour, but not the universal supremacy which he regarded as his due. The Pope viewed infallibility as his own prerogative; the Greeks held that in matters of the faith the final decision rested not with the Pope alone, but with a Council representing all the bishops of the Church.
An effort is now under way toward reconciliation.
In 2004, Pope John Paul II handed over the relics of St. John Chrysostom and St. Gregory Nazianzen to Patriarch Bartholomew as a sign of reconciliation. The relics of St. Gregory Nazienzen were brought to the Vatican during the iconoclastic controversy when the Emperor had outlawed the veneration of relics. Those of St. John Chrysostom were brought to Rome in the 13th century by the Crusaders.
The matter of authority was discussed by a joint commission of theologians in 2007 at Ravenna, Italy. In the "Ravenna Document" #41 we read, "Both sides agree that this canonical taxis [order] was recognized by all in the era of the undivided Church. Further, they agree that Rome, as the Church that presides in love according to the phrase of St. Ignatius of Antioch (To the Romans, Prologue), occupied the first place in the taxis, and that the bishop of Rome was therefore the protos [first] among the patriarchs. They disagree, however, on the interpretation of the historical evidence from this era regarding the prerogatives of the bishop of Rome as protos, a matter that was already understood in different ways in the first millennium." There is in the document a recognition of primacy at different canonical levels which is more than primacy of honor. The commission further states, "#40. History records the consultations, letters and appeals to major sees, especially to that of Rome, which vividly express the solidarity that koinonia creates."
2) "The second great difficulty was the Filioque. The dispute involved the words about the Holy Spirit in the Nicene Constantinopolitan Creed. Originally the Creed ran: 'I believe ... in the Holy Spirit, the Lord, the Giver of Life, who proceeds from the Father, who with the Father and the Son together is worshipped and together glorified.' This, the original form, is recited unchanged by the east to this day. But the west inserted an extra phrase 'and from the Son' (in Latin, Filioque), so that the Creed now reads 'who proceeds from the Father and the Son'." (website: Orthodox Christian Information Center)
The EASTERN FORMULATION begins with the pre-eternal origin of the Spirit from the Father alone.
The WESTERN FORMULATION stating point is the consubstantial communion between Father and Son.
In actuality these two perspectives on the Trinity are not opposed to each other but complement each other as stated in the Catechism of the Catholic Church #248--
Let us hope and pray that this unity may be realized soon.