I realize that your class likely is already over, but in case you remain interested in this subject, I’m going to chime in.
The best book on this subject, by far, is Sherrard’s THE GREEK EAST AND THE LATIN WEST.
The differences between the Orthodox and the Catholics are extensive. They have to do both with theology and with praxis. They also include markedly different understandings of Christian history.
The Orthodox accept the Cappadocian teaching concerning the Trinity. Notably, St. Basil’s treatise On the Holy Spirit captures the Orthodox position on the way that we are to understand the relationship among God’s hypostaseis. This is why the filioque is completely unacceptable as a theological matter.
Besides that, the Orthodox accept the Nicene position on ecclesiology–that the Creed is only to be amended via ecumenical synods. Of course, the filioque was imposed at Rome c. 800 at the insistence of the Franks and without anything like an ecumenical synod. Ultimately, the course on which the papacy embarked at that point culminated over a millennium later in Vatican I’s assertion that the pope is when speaking ex cathedra infallible. Orthodox view this as heretical insofar as it amounts to a rejection of the Nicene Fathers’ definition, which has been accepted by the Orthodox Church since AD 325 and which is consistent with the earliest experience of the Church as detailed in the Book of Acts, reflected in the Epistles of St. Ignatius of Antioch, etc.
For the Catholics, it makes sense for a pope to say “I am Tradition.” For the Orthodox, this is entirely inadmissable. Catholics insist that the pope speaking ex cathedra has never erred; Orthodox understand that the filioque was not a clarification, but a rejection of the Cappadocian position on the Trinity–and that thus it is impossible for Rome to have held both positions without ever having been mistaken.
Compounding the difficulty here is that Benedict XVI announced that Catholics may say the Creed without the filioque. If one sees the two versions of the Creed as being incommensurate, this obviously makes no sense.
As far as the Orthodox are concerned, only the Church–not any one individual or office–is infallible.
Another significant theological matter on which the Orthodox and the Catholics disagree is Original Sin. The Orthodox explain this discrepancy by reference to St. Jerome’s mistranslation of Romans 5:12 from Greek to Latin. The Orthodox have never had the Augustinian/Calvinist tradition of saying people were born guilty, unbaptized infants were damned, etc., because they did not have this mistranslation. You can find a full consideration of this matter not only in Sherrard, but in Pelikan’s volume on eastern Christianity.
In the last two centuries, Rome has been essentially following the Protestants away from Tradition, if a few decades behind. Thus, Catholics no longer follow the ancient fasting traditions–even the pious have only very, very attenuated fasts; Catholic priests now face the people during the mass, as if performing for an audience instead of leading prayer to God; Catholic religious commonly wear secular garb; Catholic temples commonly feature untraditional art; new Catholic temples typically are products of random architectural impulses, Catholic parishes commonly feature guitars, drums, etc., in their music; and on, and on. The Orthodox liturgies of St. John Chrysostom, St. Basil the Great, and St. James, on the other hand, are in their structures and chief components very ancient, as their names indicate.
Among the Russian Orthodox, there was in the 18th and 19th centuries some movement toward western-influenced church art. In the last century, however, this tendency has been seriously undercut, so that today’s new churches feature traditional Orthodox art, and much of the western-influenced art has been replaced with painting such as one would have found in the older church buildings all along. This is not an aesthetic matter merely, but a theological one. I can give you references on this topic–or indeed on any of these–if you would like.
While authority in the Catholic church comes from the pope, there is no such world prince in the Orthodox Church. Rather, autocephaly–what in American government is called “federalism”–is the guiding principle. The Ecumenical Patriarch has a primacy of honor in the Church, but only that. Besides that, the Athonite Fathers have very powerful influence in the Church, particularly but not solely among the Greek speakers of Greece, Cyprus, the Middle East, and the Western Hemisphere.
Perhaps even stranger to a Catholic would be the fact that in Orthodox Tradition, the people have a role in selecting bishops. It has not been unknown for the clergy’s choices even for the highest offices, including the Ecumenical Patriarchate, to be denied their posts because the people objected. There is a point in the ordination ceremony at which the chief celebrant asks the people to proclaim him “worthy” (Greek “axios”), and if he hears “anaxios” (“unworthy”) instead, the nominee is not enthroned.
There are of course extensive historical arguments for all of these positions and practices. In American media, one tends to hear the official Catholic position presented as the truth, even when it clearly isn’t. So, for example, television coverage of enthronements of new popes commonly features Catholic commentators calling the papacy “the world’s oldest institution,” even though of course a cursory reading of the Bible will show that the sees of Jerusalem and Antioch are older than that of Rome. Although there are Catholic sees in those cities, they date only to the Crusades; the Orthodox ones are the originals, traceable to the days when St. James presided in Jerusalem and St. Peter did in Antioch.
One hears nowadays of efforts toward a rapprochement between the EP and the pope. If these efforts are to lead to anything other than a return of the Roman see to the Nicene ecclesiological and Trinitarian positions, they will doubtless be rejected by the Athonite fathers, the Patriarch of Moscow and various other high-ranking Orthodox clergymen, and the laity. Seemingly the only point on which Rome insists is its universal authority, and so this is at root a non-starter.
One of course cannot explain this matter in full in an online comment. If you have further questions, feel free to ask. You may perhaps be interested to know that I was raised an Evangelical Protestant and converted to Orthodoxy as a result of historical investigation. That the Orthodox services, which I learned in the Greek, are so beautiful was just a bonus.