It's cultural for the last few generations. From a historical perspective it is rather inaccurate.
Let me just scratch the surface of the history of Slovakia and the Czech Republic. It is much more complex, but I hope this simplification will be enough for getting a feel for it.
Czech lands, which are east of Germany and Austria were a part of the same Holy Roman Empire for roughly 900 years until the beginning of the 19th century and then part of Austria-Hungary until the First WW. The influence is clear in trade, culture, education.
Slovakia is a bit different story. During the same 900 year period it was part of the Hungarian Kingdom, which was again a Central European entity.
Both countries sort of met in Austria-Hungary in the 19th century.
Post WWI they formed Czechoslovakia, from 1918 to 1938. It was democratic country with strong ties to the West.
The most intensive period of Eastern European/Russian influence was from the end of WWII to the Velvet Revolution in 1989. The time where the term Iron Curtain comes from.
This and the "slavic connection" with Russia are probably the sources of confusion. The languages are similar.
In the period of "romantic nationalism" of the 19th century, part of the so-called intellectual elite was intrigued by the idea of Pan-Slavism and wanted close ties with the Russian Empire. This was always more of a romantic idea that comes back every now and then. When the real decisions have been made (formation of Czechoslovakia in 1918, Velvet Revolution in 1989, joining the NATO and later the EU in 2004), they were always in a "westward" direction, in terms of both economy and diplomacy.
