Basically, how did the practice of Christianity become so different, and what were those differences?
EDIT: between the*
I can only answer your question partially, as artistic and architectural history are not my forte. The answer is: they grew apart over a long period.
The history of Byzantine Orthodoxy and Roman Catholicism is wrapped up in the history of the Roman Empire. Considering that after Theodosios I there was no real "joint" empire, and the situation rapidly went down-hill in the West, with Rome ceasing to function as the capital (replaced by Ravenna), and sacked several times (Goths under Alaric in 410, Vandals in 455). The migrations/invasions of groups such as the Goths, and later the Slavs and Avars, especially into the Balkans, geographically cut the empire in two as well.
At the same time you had a linguistic divide right from the start, with the East predominantly Greek speaking, and the west Latin. This was reflected in cultural differences as well. Those cultural differences were only exacerbated by the split in the Empire, so that the Christianities developed along different trajectories.
Thirdly you have the growing idea of the Papacy. The prominence of Rome was recognised in the East, but not always in the terms that Rome itself employed. Particularly through the 5th century you see new ideas emerge, for example with Innocent I and Boniface I (402-417 and 418-422) the idea of "vicar" applied to bishops, so that bishops apart from Rome hold their authority through the bishop of Rome, and that he in turn is the representative of Peter. Personally I would date the emergence of a full-blown concept of 'Pope' to Leo I (440-461), who consciously expresses the theory that Peter held authority over the other apostles, and transmitted that self same authority to the (next) bishop of Rome, so that the bishop of Rome is the conduit for the authority of all other bishops.
Meanwhile the theological developments in the East post-Chalcedon (the Council of Chalcedon 451) revolved around variations of Christological positions. So you get the emergence of miaphysitism in various forms that remain inside and outside the church. These are the controversies that rage through 5th and 6th century orthodoxy, morphing into the debate on whether Christ has one or two wills. During this time you have several temporary schisms between Rome and the Eastern church.
We might also note that Orthodox religion developed with a combination of eastern monasticism and classical Greek learning. The latter was fading out in the West, with Boethius really the last great Greek scholar of formidable learning. The iconoclastic controversy seems to have been partly in response to the rise of Islam and its strictures on representative art, but the reason it became a huge deal in the East has to do with the prominence of icons in Church practice there. Christian art and liturgy was developing on different lines in the West.
Meanwhile, whereas Byzantine Orthodoxy continued a very Greek-orientated culture, the West had to adapt from its Roman origins to the emergent Frankish context, its cultural moorings were headed in a different direction.
In my view the Great Schism of 1054 was the outworking of a long drift apart, that was effected because both Leo IV and Michael Cerularius were strong individuals who were willing to play hard-ball politically and theologically, and they turned what was a long-established divergence into a solidified split. Some of the theological issues included:
tl, dr: the Byzantine Orthodox and Roman Catholic churches were drifting apart for a good 500 years before the Great Schism which codified that reality.