Edit: This WAS a final question. I missed some points on this question and my professor did not specify exactly what I missed so I am just looking for more thoughts.
Edit: What I put: There were several causes of the Great Schism on both the political and religious side. Looking at it from the political side, the major change was the splitting of the Roman Empire in the 400s AD. After this split the western empire and eastern empire drifted apart and became more and more different in aspects such as politics, culture, and views of the church. When looking at it from the religious side, there were a few major disagreements between the split empire. One of the most important was the "filioque" controversy regarding the Nicene Creed. The Creed originally stated that the Holy Spirit proceeds from God the Father. The Roman church added the word "filioque" which changed the Creed to say that the Holy Spirit proceeded from the Father and the Son.
Okay, let’s go through what you put and nuance it.
The political split: While it’s true there was a ‘splitting’ of the Empire with the nominal ‘Fall’ of the Western Empire in the late 5th century, this ought to be traced back further. Firstly, you have an administrative division under Diocletian’s Tetrarchy system which, to some extent, lays the structural foundation for how the Empire will come to be regarded as ‘East’ and ‘West’. And yet even that division occurs the way it does because there are underlying cultural, and linguistic, differences between the Eastern and Western portions of the empire. In linguistic terms, the dominance of Greek in the eastern Mediterranean never gave way to Latin except in certain domains of speech (governance, law, and the military to some extent), so that throughout the whole history of the church there has been a linguistic divide through the church (within the Roman Empire).
That trajectory does lead, as you say, to ‘drifting’, especially since the western half of the Empire operates in Latin (although the church in Rome operated in Greek for a considerable time (it appears to have started switching over in the 4th century, with the liturgy changed to Latin in the 4th century), and increasingly did not participate as fully in theological discussions occurring in the East. One way to see this is the fact that of the Seven Ecumenical Councils, they all occur in the Greek-speaking east, are heavily attended by Greek-speaking Eastern bishops, and usually the Western presence involves legates or other representatives from the Pope.
Those differences are exacerbated by the political fall of the Western half of the Empire, in which the church in the West begins to operate independently of the political structures that it was used to, while the history of the Eastern church continues to be heavily intertwined with the emergent Byzantine structures.
Another factor to consider is the growing development of the idea and the claims of the Roman papacy. The growing prominence of Rome can be traced well back to the 2nd century, but the ‘idea’ of Roman supremacy develops throughout the period, with a major shift occurring in the 4th century. Damasus (366-84) began to speak of Rome as ‘the apostolic see’ and of its ‘primacy’ in association with Matthew 16:18 and develop the idea that it thus had authority, this found further development in Leo I (449-61) who particularly elaborated the theory that the Roman bishop inherited authority from Peter.
One important thing to remember about the Great Schism is that no-one really knew that it was the Great Schism until much later, because the Eastern and Western churches had fallen into schism before and repaired it. The Photian schism of the late 9th century is a good example of this, precisely in that it was really a Byzantine internal affair that the Roman Pope got involved in, it involved theological issues that would resurface later, and it was a schism that didn’t ‘last’.
The Filioque in my view is an iceberg issue – it’s small in itself, but it tears ships apart because of what lies below the surface. That is, by the time the Great Schism occurs, there are significant differences that have developed between Eastern and Western theology, including how they conceptualise the Trinity. However, the development of filioque in the West was probably meant in a way acceptable to the East, but one of the main issues was that it was simply ‘added’ without any legislative power to do so, and so the East would not accept it, while the West would not relinquish it. It reveals both an underlying theological difference, while at the same time benchmarking the refusal of both communions to cede a matter of authority to the other.
However, and this is also important, filioque did not actually play that great a role in the debate in the 11th century, but afterwards. More pressing ‘presenting’ issues involves authority of lower Italy, as well as different views on the primacy of Rome. Theologically, there were the issues of unleavened bread for the Eucharist, compulsory clergy celibacy (in the west), Whether Christ was permitted to be depicted as a Lamb, and whether one could eat strangled meat.
For more on the Filioque issue in particular, see A. Edward Siecienski’s book The Filioque: History of a Doctrinal Controversy. On the schism itself, Runciman, The Eastern Schism (1955) is old but still useful.
I haven’t addressed results of this conflict at all, that’s probably for another time (and it’s a bit out of my major time period).
Since it's that time of the year again, gotta check: Is this a homework question?