In other words, if X contains Y, do you say "X's Y"? Does the fact that X contains Y constitute possession of Y by X? "Russian regions" is 10 times more popular on Google Scholar, so "Russian" is probably correct, but I wonder if there is a concrete rule that can be applied in cases like that.
Both of those could be used and be grammatically correct. The difference is slight, though. Essentially, "Russia's regions" refers to the regions contained/owned by Russia, while "Russian regions" refers to regions which, while not necessarily contained/owned by Russia, are still in some way connected to Russia.
The difference is that of a possessive and an adjective. You would use 'Russian' in the same kinds of places you would use 'Cold' or 'Slavic' because they describe the region. Even places that aren't actually part of Russia, like the ethnically Russian part of New York, could be described as 'Russian regions.'
"Russia's regions," in contrast, describes only regions owned by Russia. Regions that couldn't be described as 'Russian' except for the fact that Russia owns them would definitely fall under this category.
All in all, though, there is a lot of overlap between the two categories, and native speakers would understand easily if one mixed them up.