This divide is because much of the Eastern half of the Empire consisted of the Hellenistic lands that had been ruled over by the Diadochi kingdoms- the successors to Alexander. Basically it's a huge arc from Greece itself, all the way through Anatolia, the Levant, Egypt and Cyrenacia where Greek was the lingua franca, and the prestige language. In the Eastern Mediterranean, even where the locals would have spoken a different language among themselves (e.g. Aramaic in Judea, Egyptian in much of Egypt), the language of the urban and mercantile classes was Greek (which is why the Apostle Paul, for example, wrote his epistles in Greek despite being a Jew). When the Romans conquered the Diadochi lands, they were conquering rich, highly organised and sophisticated societies with a certain level of cultural prestige. The Romans slotted their administration in at the top but there was little "Romanisation" of these regions. The Romans themselves viewed Greek culture as prestigious on some levels. They had complex and varying attitude toward it- on one level they saw the Greeks as effete and decadent, on the other they respected and valued the philosophical and cultural achievements of Greece, and well born Roman boys generally had Greek private tutors to teach them at least some of the classics. The seeming similarities between the two cultures are more of a modern conflation, really. I suppose a modern analogy would be for someone to say that British and American culture are essentially the same. To a non-Anglophone, they might seem to have more in common than not but any American or Brit can probably point out more differences than commonalities.

Anyway, in the Eastern Mediterranean, there wasn't that much of a challenge to Hellenistic culture as the prestige culture in contrast to the Western Empire. Here Rome's culture became the prestige culture. Gauls, Britons and the like adopted Roman cultural practices and Latin was the lingua franca.