The Great Schism, or the schism between the Roman Catholic Church and the Eastern Orthodox Church, is a complex issue that started well before the mutual ex-communications in 1054 CE. Before we get into the theological, cultural, and political differences that led to the schism, we should note that there is a nuance to understand how it happened. The schism was a long process that was not completed until well into the seventeenth century in some parts of the world.
The roots of the Great Schism trace themselves back to the council of Nicea in 325 CE, when the Nicene Creed was formulated as a defense against the teachings of Arius. The creed became an essential part of the Catholic faith and was reaffirmed at later councils. In 381 CE, the council of Constantinople took place and added additional clauses to the creed concerning the Holy Spirit.
One specific clause, “And in the Holy Spirit, the Lord and giver of life, who proceeds from the Father,” became the subject of controversy when it was altered in the West. This clause would be one of the central theological issues contended with among the bishops of the Catholic church. However, many other problems arose between the East and West that led to the Schism in 1054 CE—but none was as contentious as the Filioque clause.
As time went on from 1054 CE, the theological differences compounded, but the critical theological difference was the adoption of the Filioque clause in the West. The Filioque finds its origin in Spain at the Third Council of Toledo. The council was primarily focused on defending the faith against Arianism and inserted the clause as a means to add authority to the claim that Christ is God. The Filioque clause adds the words “and the son” to the earlier clause so that it reads, “And in the Holy Spirit, the Lord and giver of life, who proceeds from the Father and the Son.”
The phrasing of the Filioque became common among Latin Christians in the West—but was rejected by bishops in the East. The issue came to a head in the ninth century when there was a minor schism over the issue, called the Photian Schism. However, the creed was not officially altered during the mass in Rome until 1018 at the behest of the Holy Roman Emporer—who wanted to assert dominance over the Eastern Roman Emperor.
The Roman inclusion of the Filioque and the Eastern rejection of it provided fertile ground for the schism 40 years later. The dogmatic addition of the Filioque has been a point of contention ever since, and it is still rejected by every eastern church—including the Eastern Catholic Churches who are in communion with Rome.
Also, a minor theological issue was the difference in the bread used in communion. The West used unleavened bread, while the East used leavened bread. It would not have been a big issue, except that it became a point of contention for the Pope.
Ecclesiology became a contentious issue leading up to the great schism. Ecclesiology is how a church is structured and governed. There were two principle disagreements among the Eastern and Western parts of the church. The first was the issue of primacy and the second the issue of celibacy.
The primacy of the Pope of Rome became a contentious issue in the ninth century, and tensions continued to be high up until the schism in 1054 CE. The Roman Catholic Church contended that the See of Peter, or Rome, was granted exclusive supreme jurisdiction over all other churches. This idea seemed novel to the Eastern churches, and every Patriarch in the East rejected it. Eastern ecclesiology was based on the idea that every bishop had jurisdiction over his parish and that councils and synods held authoritative power over the whole Church—not a single bishop. Eventually, because the Patriarch of Constantinople would not follow the decisions and declarations of Rome, the writ of ex-communication was placed on the altar of the Hagia Sophia.
Along with the issue of who was in charge, there was an issue of whether or not married men could be made priests. In the East, the only clerical office that required celibacy was the bishop. A married man in the East had always been able to be ordained to the priesthood, and that became a point of contention between the bishops of the Latin and Eastern churches. While not a dogmatic issue, an issue of faith, it was made another issue that showed the East’s defiance of the Pope.
The principal head of state in the East was the Eastern Roman Emperor, while the West, after the fall of Rome, was the Holy Roman Emperor. This led to tensions as the HRE looked to assert itself as the major Christian political power. The Eastern Roman Emperor was on the back foot because of the ever-looming threat of Islamic invasion that continued to push upward through the empire—but political differences created a strain on the religious institutions of the East and West and helped to deteriorate their relationships.
On a side note, while the sacking of Constantinople happened in 1204 CE, it cemented the schism and made it permanent. In the sacking, the crusaders, who were meant to help the Greeks, instead destroyed the city by raping, pillaging, and stealing many religious artifacts. The Empire and Eastern Christians never forgot the atrocities, and it put to rest any hope of eventual reconciliation.
The cultures of the Greeks and Latins contributed to issues in the schism. Many of the Greek bishops couldn’t read Latin and the Roman Bishops couldn’t read Greek. This created a myriad of problems when theological or political differences surfaced—because dialogue was impossible, and translation could lead to confusion.
The inability to understand language also led to further difference because the major saints and theologians took different stances that diverged over time, leading to greater confusion and issues in councils and synods.
There are two examples of the Great Schism. The first example occurred in 1054. In this instance, the Byzantine Church split with the Roman Catholic Church. In this situation, the representatives from both churches excommunicated the leader from the other church. A representative from the Roman Pope excommunicated the patriarch of Constantinople. As a result, the patriarch of Constantinople excommunicated the Roman Pope. This led to the creation of the Eastern Orthodox Church (located in Constantinople) and the Western Catholic Church (located in Rome). There were differences between each new church. The Eastern Orthodox Church used Greek as the language while the Western Catholic Church used Latin as the language. This led to communication issues and differences between the services of each church.
Another schism was Western Schism. In this situation, two people claimed to be the official pope. One pope was based in Rome, and the other pope was based in France in the city of Avignon. As a result, the leaders of Western European countries had to declare loyalty to the Pope of Rome or to the Pope of Avignon.
Actually, the differences were much deeper than those indicated above. The Western Church, considered itself to be the one true church, the universal, hence "Catholic" Church. The Church in Constantinople considered itself the correct and traditional church, hence the "Orthodox" church.
The people of the Western Empire spoke Latin, and church services were conducted in that language. The people of Constantinople and the Eastern Empire spoke Greek. The Latins considered the people of Constantinople to be effeminate and unmanly; the people of the East considered the Western people as barbarian and classless.
A further issue dividing the two was the use of Icons. The Eastern Church considered them graven images and thus a violation of the Second Commandment. The Western Church saw no such problem and used Icons of saints and the Blessed Virgin as well as Crucifixes regularly.
Finally, there was the issue of the supremacy of the Pope. The Pope originally was one of four patriarchs of the Church, one in Rome, Constantinople, Alexandria and Jerusalem, respectively. Alexandria and Jerusalem fell to the Muslims; after which the Pope claimed to be the successor to St. Peter and the Vicar of Christ on Earth. The Eastern patriarch did not agree; and each excommunicated the other. The schism thus became permanent in 1054.
I assume that you are asking about the schism of the Christian Church into the Roman Catholic and Eastern Orthodox churches. This schism happened gradually over centuries and became official in the year 1054. It had political and religious causes.
The political cause was the splitting of the Roman Empire. In the 400s AD, the Roman Empire split into a western empire (capital at Rome) and an eastern empire (capital at Constantinople). The two empires became more and more different in terms of their politics and culture and their churches grew apart as well.
On the religious side, there were many doctrinal disagreements. Perhaps the most important of these was the "filioque"controversy over the Nicene Creed (a statement of the core beliefs of the faith). The Creed originally said that the Holy Spirit proceeds from God the Father. The Roman church came to add the word "filioque" which changed the Creed to say that the Holy Spirit proceeded from the Father and the Son.
These were the major reasons for the Schism.