Saint John Paul II spoke of Christendom having two lungs: the Western and Eastern Churches represented by Rome and Constantinople. He said the Church “must breathe with” both (rf. Ut Unum Sunt, n. 54).
Many Eastern Christians have been readmitted into the Church after almost 1000 years of schism, but many still remain outside of full communion with the Church. Those are the Orthodox. The Eastern Orthodox, like us Roman and Eastern Catholics, are Chalcedonians, meaning they profess faith in the hypostatic union of the Jesus Christ’s divinity and humanity as two natures in one Person. Interestingly enough Catholic apologists have noted the schism was not formal until the 15th century after many Byzantines refused to accept the terms of the Council of Florence regarding the Filioque, the use of unleavened bread for consecration, and the dating of major feast days.
Many of these Eastern Christians accepted the terms of the Council of Florence and were admitted into full communion with the Roman Church mostly in the early modern period. They accepted that the Filioque is a legitimate expression of the Nicene Creed, that the use of unleavened bread does constitute for a valid Eucharist, have accepted doctrines which were later defined as dogma such as the Immaculate Conception, the Assumption and papal infallibility.
These dogmas, divinely revealed truths, are stumbling blocks for the Orthodox in their reconciliation with the Holy See, especially papal infallibility and papal primacy. They say the pope’s scope of power went too far somewhere in the Middle Ages. Initially this was not an issue for the Orthodox as the Great Schism first started as an issue between the Normans and the Byzantines. The Normans conquered Byzantine Sicily and soon afterwards were demanding priests to say Holy Mass in Latin, not Greek. The patriarch of Constantinople did not like this, so he demanded priests Constantinople to say Holy Mass in Greek. Eventually three papal legates were sent to give a bull of excommunication to the patriarch who in turn excommunicated the legates, not the pope who had already passed away.
Eventually there was the massacre of the Latins and the sacking of Constantinople which for a short time made relations between the two bitter. There was bad on both sides and neither one was exactly innocent. Catholics and Orthodox have learned that we share the same faith albeit are not yet in full communion. We have learned especially through hardships with Protestants who think the same about the Orthodox as Catholics, not to mention the Islamic persecution of Catholics and Orthodox, that Catholics and Orthodox have a lot more in common than what divides us.
This is unlike anything Catholics and Protestants share: Protestantism is heresy, Orthodoxy is not. Protestantism has no valid priesthood and therefore no valid Eucharist, Confession, Holy Orders, Confirmation, or Anointing of the Sick. Orthodoxy does have all seven Sacraments and the Orthodox too believe there are seven Sacraments, no more, no less. The similarity makes it pretty easy for Catholics and Orthodox to dialogue, not so much for Catholics and Protestants. We must hope and pray more Orthodox will come into full communion with us.