I don't just mean this because "western" essentially means white, but they seem more culturally alike to Europe and America than to countries like Japan, China, Persia, India, etc.
Their country was a Marxist republic for a while, Marx was a man of western philosophy no doubt.
What makes them not consider themselves apart of "the west"?
They have historically not been close to the west, stretching back to their domination by the Mongols in the 1200s. The result was some 300 years of rule by the Mongols in the form of the Golden Horde, which kept Russia distinct culturally from Western Europe (also the split of the Orthodox and Catholic Churches played a role as well). This meant that for most of this time Russia was more oriented east rather than west, and languished behind in development in contrast to other European states.
Some attempt was made by early Tsars to align Russia closer with the west: Ivan IV the Terrible corresponded with Queen Elizabeth I of England, for example, for years, and even had the English set up a trading post in a port (I want to say up in the Baltic region near where St. Petersburg would be founded, but I can't say for sure), but nothing ultimately came of that; Peter the Great also famously went on a tour across Western Europe that lasted years, learning how to modernise Russia and make it more like the other European states.
Furthermore, when Russia was no longer under Mongol rule they expanded to the east, not west. Thus they kept closer contact with eastern powers rather than western ones for a while (they did some western expansion, notably into the Baltics, where they fought the Swedes and Poles, and Ukraine, where the Ottoman and other Turkic powers were).
This all contributed to Russia having a distinct identity than the west. The Revolution and resulting Cold War only further cemented that divide, right through to the modern day.
Hi! hopefully this post will attract more responses, but I didn't want you to miss this great thread about the term "West" from last year:
What is 'The West'? 'The East'? Do you consider them useful terms?
I recall another related discussion in one of the stickied posts from around the same time period (probably a Theory Thursday), so if someone remembers/finds that post, kindly link it here - thx!
two things in Russian history come to my mind as being important to distinguish Russia from "The West" by which I mean western Europe and the USA. In the 20th century during the Cold War, both sides wanted to be very distinctly different cultural entities from the other one. Though there were always overtures towards peace and the avoidance of world war 3 (The Kitchen Sink Debate) the governments of the USA and of the USSR both pushed propaganda at their people making sure everyone knew that the other side's people were "different". On the Russian end of things it meant distinguishing themselves from "The West".
Of course, this was related to a deeper-seeded Russian culture war. During the reign of Peter The Great from 1682 to 1721, he (especially when he first took power) made many attempts towards modernizing Russia including making men shave their beards, reorganizing the army and generally pointing towards "The West" as an ideal towards which Russians should strive to imitate. This was met with not some small level of resentment. In fact, after it and right up until the Russian Revolution and the deaths of the last Romanovs, "The West" was viewed as something that the nobles wanted and that was fundamentally different from the proletariat. Lenin and company capitalized on this while also seeing the necessity of adapting western industrialist policies.
Sources: Walter Lafeber (2002). America, Russia, and the Cold War, 1945–2002.
James Cracraft. The Revolution of Peter the Great (Harvard University Press, 2003)