I’m reading an old textbook on European society and it says that the split between West and East civilization can be traced to Ancient Greece who saw the older civilizations they grew up under as “Eastern” and wanted to define themselves as something different. According to the book, they later bring this idea to the Romans. I’m wondering though, why the ancient Greeks wanted to separate anyway? It doesn’t say in any of the pages WHY Ancient Greece decided to do that in the first place. Was it just to set about their own legacy as their own civilization and didn’t want to keep being shadowed by the civilizations who came before?
Which book is this? As far as I know, there is no reason for why the Greeks would want to distinguish themselves specifically from people in the Near East: it's good to remember that, from at least the fifth century BC, the Greeks distinguished between themselves and the barbaroi, that is, "speakers of bar-bar languages" (i.e. non-Greeks, not "barbarians" in the sense of rough, uncivilized folk). Any notion that the Greeks deliberately did something different after e.g. the Orientalizing phase in their art history (7th century BC), which is what I'm sort of getting is what the book you read suggested, is simply teleological.
In any event, there's this idea of a great east/west dichotomy, where the Greeks are European/rational/democratic and the civilizations of the Near East are Eastern/irrational/despotic. Any scheme that is this clearcut is immediately suspect. Concerning this notion, I gave a detailed answer on AskHistorians here that may help you.
A pertinent quote from that reply of mine:
In practice, it’s difficult to pinpoint where ‘Aegean/Greek culture’ ends and the ‘Near East’ begins. There is a lot of overlap. For the historical period, for example, it’s clear that many Greek cities in Asia Minor and on the islands off the coast had closer ties with Anatolia than with mainland Greece, such as Lesbos: see e.g. A. Dale, ‘Alcaeus on the career of Myrsilos: Greeks, Lydians and Luwians at the East Aegean-West Anatolian interface’, Journal of Hellenic Studies 131 (2011), esp. pp. 21–23 and 23 n. 43. In ancient Greek art history, material culture of the seventh century BC is so heavily influenced by Near-Eastern art that we speak of an ‘Orientalizing’ period.
Other users -- /u/Iphikrates and /u/Iguana_on_a_stick -- have tackled the same issue here as replies to a question on the supposed difference between Eastern and Western modes of warfare.