The separation between what would later become the Roman Catholic Church and the Eastern Orthodox Church is what we now call the Great Schism of 1054.
There have actually been many theological and ecclesiastical differences between the two factions owing to cultural differences over the centuries, but it was only in 1054 when those differences finally became too great to overcome and separation became formal.
It was the split of the Roman Empire in 284 CE into the Western Roman Empire and the Eastern Roman Empire that eventually led to the separation of the churches. After the Eastern Roman Empire (Byzantium) became on its own, the bishop of Constantinople (the capital of Byzantium) was declared second in honor to the bishop of Rome (the Pope).
Over the years the influence of the bishop of Byzantium would also grow and the two churches would clash over theological and ecclesiastical differences like whether or not unleavened bread should be used during communion, the Quartodecimanism issue, the human and divine nature of Jesus Christ.
They also disagreed on the role of the Archdiocese of Constantinople in Christendom, the concept of original sin, and the existence of purgatory.
As early as 1053 misunderstandings were already becoming irreconcilable. Churches in southern Italy sympathetic to Constantinople were ordered to conform to Roman practices or close. In the East, Latin Churches were likewise ordered to follow the teachings of the bishop of Constantinople or also face closure.
Eventually, the bishop of Constantinople at the time, Michael I Cerularius, would ask Pope Leo IX to grant him the title Ecumenical Patriarch to distinguish himself from the Pope.
The Pope sent a legate led by Cardinal Humbert of Silva Candida to officially refuse the title to Cerularius and insist he submit to Rome’s authority. Cerularius refused and was excommunicated by Cardinal Humber. In turn, Cerularius excommunicated the cardinal.
Quartodecimanism refers to the controversy of celebrating certain religious events, particularly The Lord’s Supper and Easter, between the churches in Jerusalem and the churches in Rome. The problem happened because days in Europe “began” during sunrise, but in the areas around Jerusalem the day “began” after sunset.
Both have practically the same beliefs regarding God the Father, Jesus, and the Holy Spirit. The main differences lie in the role of the Papacy, how their followers hold worship, and how they live their lives as Christians.
His Holiness Pope Francis has served as head of this church since 2013.
His All Holiness Bartholomew I has served as Ecumenical Patriarch of Constantinople since 1991.
While Rome constantly reaches out to the Orthodox churches, the Eastern Orthodox Church said it isn’t interested in a reunion at this time.