The crucial period between the 6th and 12th centuries AD, that is, for a full 600 years, there was no civilization in the entire world that struggled with entropy and preserved the ancient Greek historiography, the Homeric epics, and the Ancient Greek theater. In the Carolingian West, only the works of the Christian fathers such as Dionysius the Areopagite, Gregory of Nyssa, etc. were translated. There were also indications of translations of Plato and Aristotle, but that was it.
At the same time, in the Arab world, the translation of Greek works was more widespread, but even there selective. Scientific and philosophical texts of Galen, Hippocrates, Ptolemy etc. and philosophical texts such as those of Aristotle and Plato. One may ask, since there were already translations of ancient texts, what important did the Romans (Byzantines) save in these 600 years or so?
Imagine a world where there is not a single play by Euripides, Aristophanes and Aeschylus and not a modern performance of them, no book, no inspiration from their works. Imagine a world where we know very little about the Persian Wars, the Marathoners, the Battle of Salamis, the 300 of Leonidas, and whatever other historical information Herodotus gave us. Imagine a world where we only have clues about the Peloponnesian War and not the details and of course the writing skill of Thucydides. Imagine a world without the Iliad and the Odyssey.
This was certainly not a world dreamed by the Byzantine civilization (a common argument of detractors of this period). Even in extreme times with the religious strife of iconoclasm, constant raids, epidemics, etc., New Rome preserved the memory not only of the ancient Greek world but also of the entire Western civilization.