What led to the fundamental split of the Roman Catholic and Greek Orthodox churches?
There were two main factors that led to the schism between what we now call the Roman Catholic Church and the Greek Orthodox Church. One set of factors was political while the other was religious.
One part of the schism came about because of politics. There was political competition between the Byzantine emperors and rulers in the West. Both wanted to increase their power and prestige. This led them to support the idea that the church should be run by people from their region. This sort of tension started to pull the Byzantine church away from the Western church.
The other part of the schism came about because of religious differences. By the 1000s, a large number of differences had arisen between the two churches. They had different stances on clergy being married. They had different rules about fasting. They used different languages in their liturgies, with the Roman Catholic Church using Latin while the Orthodox used Greek but also allowed non-Greek speakers to use their vernacular languages. They had differences in their beliefs about papal authority and about the nature of the Trinity (the filioque controversy). Finally, there was the iconoclasm controversy in which Byzantine Emperor Leo III tried to outlaw the use of icons (religious images). This caused conflicts with the West where the popes did not agree that the use of icons amounted to the worship of idols. These differences in belief and practice pulled the two sides farther apart.
Thus, the schism which took its final form in 1054 had its roots in both political and religious differences.