So before the Roman Empire included the east, Greek was the “lingua franca” of the Eastern Mediterranean and Asia Minor, and because the Roman elite respected Greek as a philosophical and literary language, they left it alone in the East and indeed made it one of the two official languages of the Empire. So for a Greek-speaking resident of the East, you could still continue your day-to-day life and even interact with the Roman state, all without having to learn Latin.
The same factors weren’t really there in the west, where there were a lot of different languages, none of which had the prestige and thus utility of Greek. And while the Roman Empire didn’t try to impose Latin on conquered people in the West, because Latin was the only language of administration, law, military, and business, there was an incredibly strong incentive for the children of the provincal elite to be educated in Latin so that they could become Romanized and advance in society.
That being said, it should be noted that bilingualism in elite circles was the rule in both the West and East through the 5th Century CE. Indeed, for a long time, because Latin was considered the “language of power,” there were complaints that the study of Latin by high-status residents of the Eastern Empire was harming the quality of traditional education in Greek rhetoric.
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