You have to go way back in history to answer this one with anything approaching accuracy. The distinction wasn’t between Europe and Asia or eastern Europe and western Europe. It was originally between Latin (western) Christianity and Greek (eastern) Christianity.
It starts with the Roman Empire. Rome was divided into two broadly separate administrative units that ultimately became two separate empires – these were split between the west and the east, first administratively, later politically, and finally linguistically and culturally. Every (even American) school child probably knows the Western Roman Empire nosedived in the 400s and ultimately fell. What usually isn’t driven home is that the eastern Empire lasted another thousand or so years, until 1453 – to put that in perspective, Christopher Columbus was already alive and the modern world was starting.
So, after the Western Empire fell (officially in 476, but putting an official date on it probably doesn’t make much sense), Germanic tribal warlords took over most of territory that the empire occupied – and other than a few attempts to reclaim the western territories, or at least Italy, by the Eastern Empire, it never really was meaningfully politically unified again. The most culturally cohesive force for this part of the former Empire was the Roman Church (Roman here means Rome the city). Bishops and priests were probably the only people educated enough to at least kind of administer a territory, probably while taking orders from whatever Germaic warlord was in charge. For several centuries, the office that is now called The Papacy was at least in theory a co-equal religious patriarch with other such patriarchs in the Eastern Empire, such as the Ecumenical Patriarchate of Constantinople, and all were nominally subordinate to the Roman Emperor still sitting in Constantinople.
Over time, for a number of reasons, the western church became politically and culturally more independent. One reason is the patriarchs in the east tended to maintain Greek cultural ties, while Rome stayed Latin. Linguistics is another – the Eastern Roman Empire dropped Latin as its language in the late first millennium and became Greek-speaking. Geographically, the economically less powerful west was stuck between broadly the Eastern Empire and its client states and Muslims invading from Spain and North Africa, while the Eastern Empire was acting as a buffer to keep invading Muslims out of the west.
By the time of Charlemagne, the Pope managed to consolidate much of his power in the west. Roman Catholic missions to the north and west managed to bring heathen western Germans and Celts into the fold – this is why these countries are now considered western, even though they usually weren’t even part of the empire and ultimately ceased to be Catholic in some cases. Meanwhile, Eastern Christian (broadly speaking, we can say Eastern Orthodox) missions went north and somewhat west to countries like Russia and Romania – so these countries became Orthodox (eastern).
Eventually, of course, Muslims managed to conquer the almost the entirety of what was the Eastern Roman Empire – we call this the Byzantine Empire today. This includes what is today Turkey, North Africa, and some of the so-called “Middle East,” no longer Eastern Christian.
You see some of these cultural distinctions in other ways too: western countries use the Latin alphabet, while eastern ones stuck closer to Greek. Even languages like Coptic use at least some Greek letters. Holidays in Latin countries are a little different because of the Gregorian calendar. You also see some odd exceptions; Poland became really Catholic later on, while probably culturally being more like other Slavic countries.
Admittedly, too, the distinction between east and west has taken on different meanings over time, but the main point of being “western” is probably being part of a society that has its roots in Latin Christianity – what we today call Roman Catholic. Much of the Americas are cuultural offshoots of places that were at least once dominantly Latin Christian, while Russia’s cultural and religious identity lies more with its associations to the east.