What events led the separation of the Churches and the branching into different denominations? And what led to the difference in the attire of the priests, e.g., Orthodox Patriarchs tend to have more ornate and richly decorated robes than their Catholic counterparts?
I will not be able to answer all of these questions thoroughly but I do have some intimate knowledge of the schism between the Eastern Orthodox and Roman Catholic Churches being Orthodox myself.
The Catholic Church and Orthodox church developed differently. Part of that is due to geography, and I believe another major influence of that development was the fact that while the Eastern Church remained in a mostly politically stable environment with the Eastern Empire still mostly intact, the Catholic Church lived in a much more active and less stable Western Europe.
The churches had many differences between each other. Language, whether priests could marry, the use of icons, and many others.
The straw that broke the camel's back was recognition of the church in Rome as the mother of all churches and the Pope's supreme authority over the rest of the clergy. The Orthodox church was not willing to recognize the supreme authority of the pope.