Most of this comes from the perspective of history as written by the historians. Let's go back to the split of the Roman Empire into its Eastern and Western parts. The emperor Diocletian responded to a series of crises in the empire by splitting it into two administrative portions starting in the year 286, with each portion to be ruled over by a senior Augustus with a a junior Caesar to serve as his aide and presumptive heir. The division put what is now Britain, Italy, France, Spain, parts of Germany, and some of northern Africa into the western Empire, and modern Greece, the Balkans, Turkey, Egypt, and the Near East into the eastern Empire. For the most part, residents of the west spoke Latin and those in the east spoke Greek, although Latin was still the official language of the whole empire.

The new system worked to a limited extent for a while, but conflicts inevitably arose between the 4 men serving as Augusti and Caesars, with a series of civil wars, depositions, retirements, unretirements, and banishments following.

The mess was eventually sorted out by Constantine the Great, who assumed sole rule in 312. He desired a new capital, one more defensible than Rome among other things, and founded the city of Constantinople, which became the new capital in 330. Things began to go south again after Constantine's death, and the empire again began to be divided into eastern and western parts. The west fared much more poorly, suffering a series of invasions from Germanic and other tribes, a gradual weakening of the military, and massive losses of territory to invaders. The date traditionally given for the fall of Rome is in 476, when the last "emperor" (a powerless figurehead by that point) was deposed by a Germanic king and not replaced. But by that point, the former areas of the western empire like France and Spain had been under Germanic rule for decades. Historians traditionally set the fall of Rome as the beginning of the Dark Ages.

Meanwhile, things were going much better in the east. The empire there suffered some setbacks and lost territory, but remained intact. Here's the key point - the people there thought of themselves as Romans, even if the city of Rome was no longer under their control. Their soldiers would have told you that they served in the Roman army, and the emperor traced his power in a direct line back to Augustus Caesar. The eastern empire was even able to briefly reconquer Italy under the emperor Justinian, but were unable to hold it for long.

So why don't we (and a lot of traditional historians) thing this way?

There was a language, cultural, and religious divide that developed. The eastern emperors, like their citizens, eventually settled on Greek as the official language. Culturally, the east disdained the successor kingdoms in western Europe, which they regarded as little more than barbaric. (The term "Frank" remained used as an insult there until Constantinople fell). And possibly the biggest blow to harmony occurred in 1054 when the east rejected the religious primacy of the Pope in Rome and split away from the Church in the west.

The east, or what became known as the Byzantine Empire in the west, remained a powerful force through most of the middle ages. Constantinople itself didn't fall until 1453, a date that some historians use to mark the end of the Middle Ages. During that time, the eastern empire served as a center of learning and culture, as sat as a bulwark that stemmed the tide of conquest from the forces of Islam.

This is a very roundabout answer to your question, but the answer depends on who you ask. A historian writing in, for instance, France in 1200 or so would have talked about the fall of Rome and the slow recovery from darkness culminating int he coronation of Charlemagne. This viewpoint would echo through western Europe and be the one taught to most schoolchildren.

But from a different point of view, Rome didn't fall at all, not for another 1000 years anyway. It merely moved to Constantinople. A historian there would have given you a much different perspective. It just isn't one that is taught in the west, for the most part.

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One very simple explanation is that in the popular imagination, as well as in the eyes of some past historians, (Most notably Gibbon, of The Decline and Fall of the Roman Empire fame) the Western Roman Empire was the "real" Roman Empire, and the east was a mere rump state that shambled along until its inevitable collapse. This has fallen out of favor, as more recent histories have pointed to the fact that the eastern empire remained a significant power for about seven hundred years after the fall of the west, and survived until 1453. That is, I should note, LONGER than the western empire, which lasted for about 500 years after the overthrow of the Republic. Still, the Western Empire was predominately Latin speaking and had the small advantage of actually containing Rome, so it tends to get seen as somehow more legitimate.

This sort of outlook, again, holds more weight in the popular imagination than in the modern historical community, but it did hold more sway when a lot of these periodizations were set down. However, there are also some more historically legitimate reasons for setting the barrier about where it is: In the west, at least, the collapse of the Roman Empire cause substantial societal change: Trade links collapsed, sending the large, import dependent cities of antiquity into decline. Systems of taxation essentially disappeared, along with the elaborate Roman bureaucracy. Some of the new Germanic kings (Most notably the Ostrogoths of Italy) attempted to co-op Roman institutions, but these efforts had mixed success. No matter how hard anyone tried, there just wasn't the economic strength and cohesion within the shrunk and more isolated Post-Roman territories in the west to sustain a state that operated on the scale Rome did.

We see a huge drop off in intellectual activity and textual production at this time as well, as the need and funding for schools both dried up. While Latin literacy declined, the greater loss was the almost complete loss of Greek literacy in the west. Many ancient Greek texts had never been translated into Latin by the fifth century, as any Roman educated enough to read them would have had an education in Greek anyway. Monasteries and other clergymen would preserve many texts, and translate and compile others, but much was lost, at least to the west.

I want to be careful of pushing this decline narrative too far though, because for the vast majority of people, there would have been little change. From my understanding (And I'm pulling this from Medieval histories mostly, so Roman historians, please call me on this if incorrect) the Roman Empire left a comparatively limited mark on the countryside. Its fall might have even represented an improvement to the lot of some, as disappearing taxes might have increased local surplus of food.

So while these periodizations are to some extent arbitrary, they are by no means trivial. While placing the end of antiquity at 476 is arbitrary, it is clear that between 400 AD and 600 AD, the West of Europe went through some pretty serious changes in society and organization.

What about the Byzantines? Well, basically take everything I said before, and just forget about it. The Eastern Roman Empire was by far the most complex state in Europe, and after the fall of the Western Empire, no European empire would match its organization and centralization for centuries to come.

By the sixth century, the Eastern Empire recovered its footing fully, and drawing on the tremendous tax revenue the government could bring to bear, went on an ambitious campaign to reconquer much of the old Roman territory under Justinian. These gains would prove ephemeral, but it's important not to post-hoc dismiss them as flights of fancy. Large swaths of the empire had fallen out of Roman hands and been retaken before, and the Byzantine empire was by far the most powerful state in Europe at the time. Byzantine power would ebb and flow over the next centuries, but until the Crusader sack of Constantinople in 1204, it remained a substantial power in Europe. Even after losing much of its territorial extent, the complexity of its government allowed it to maintain a far stronger presence than any other European state of similar size.

Many texts were preserved in Byzantine libraries, and Byzantine art and architecture remained some of the finest in Europe until the fall of the empire. Unfortunately, my grasp on the philosophy and literature of the time is limited. Medieval Lit training at my school and many of the ones I know of is focused almost exclusively on Western European Lit, and the same is true, as I understand it, in Philosophy. I hope to remedy this soon, but it's hard to find English language sources. Unfortunately, the medieval curriculum most people get exposed to remains vastly over focused on western Europe, which, I think, is what makes this question necessary

In conclusion, though, Byzantium made a serious effort at preserving the Roman state in the west, and kept the Imperial flame alive in the East. It went through changes in the Medieval Period as well, but these map poorly onto the traditional categorizations of the period, which are much more focused on Western Europe. Byzantium participated in the affairs of Europe, obviously, but perhaps an even more interesting way it "plays a role" is in complicating our narratives of what the Middle Ages were and were not. It was, after all, a classical nation that stubbornly refused to Decline and Fall in a neat and even fashion.

Sources:

Colish, Marcia L. Medieval Foundations of the Western Intellectual Tradition, 400-1400. New Haven: Yale Univesity Press, 1997. Print. Yale Intellectual History of the West.

Herrin, Judith. Byzantium: The Surprising Life of a Medieval Empire. Princeton: Princeton University Press, 2009. Print.

Hollister, C. Warren. The Making of England 55 B.C.-1399. New York: Barnes & Noble Books, 1994. Print.

Wickham, Chris. Medieval Europe. New Haven: Yale University Press, 2016. Print.