[note: This post was written in response to a question posted at Quora.]
This question assumes that there was ever a distinction between the Greek world and the Roman world. In fact, the two shared a common tradition, established by the Greeks in the fifth century BCE, when the city-state of Athens invented direct democracy and defeated the most powerful empire of its time. The Greeks also invented or contributed significantly to history, philosophy, oratory, tragic drama, epic poetry, lyric poetry, sculpture of the human body, comic theater, mathematics, astronomy, and education.
The list of Greeks who contributed to this great upwelling of culture included Homer, Plato, Aristotle, Themistocles, Herodotus, Socrates, Aeschylus, Sophocles, Euripides, Pythagoras, and many more. After democracy died in Athens, Greek culture spread all over Europe, Africa, and Asia as far as India. At the death of Alexander in 323 BCE, a Greek world could be said to exist where Greek culture, language, and ideas were studied and admired.
While Greek culture was astonishing the world, Rome was a small city-state on the Italian peninsula. It was by no means the largest or most important of the small states that surrounded it. Rome finally became a regional power in 341 BCE when its armies defeated the kingdom of Carthage, which held numerousterritories in Africa and the Island of Sicily. Although Rome was conquering territories and expanding its borders, its culture was negligible until its poets and scholars began copying Greek models and studying the Greek language.
The great flourishing of Roman culture came as its armies were conquering the rest of Europe. Its aristocracy learned Greek and conversed in that language. Its authors copied Greek models. Vergil wrote the Aeneid in Latin, modeled after Homer's Odyssey. Horace wrote his Odes modeled after the odes of Alcaeus, a Greek poet. Cicero, the great Roman statesman, modeled his speech against Catiline after the speeches that Demosthenes made against King Phillip, Alexander's father. The Roman poet, Ovid, used Greek mythology as his subject matter in the Metamorphosis. Many other examples could be given, since Greece had created the only body of literature and learning in the European world at that time.
Roman legions put down an uprising by a league of Greek cities and destroyed the city-state of Corinth in 146 BCE. This was a very one-sided war that led to the reorganization of Greece into two Roman provinces. At that time, the Romans were on their way to world domination, while the Greeks were a collection of city-states that could not compete with the Roman army.
So there was never a Roman world in opposition to a Greek world.