I would argue that the USSR was far more concerned about not antagonizing the US than the Russia of today: Before it quickly escalated and turned into a major spectacle, used to demonize and vilify if not to justify military activities (every time the USSR did anything that played poorly in the US press, you could hear the military contractors cheer at the new destroyer and fighter aircraft orders incoming), whereas now there are attempts by both sides to brush it under the table. As an aside, Russia has negligible trade with the United States.
Russia stills flies nuclear bombers to the edge of US and Canada airspace. It still has agents throughout the West, many who have built lives here. The leadership in Russia is still emboldened and lauded for standing up to the US in various ways.
In many, many ways it is the same as it always was.
The only reason I think Russia might not care much about Snowden's info is the fact that Snowden got the info in the first place -- if a 6 month outside contractor had the keys to the palace, there's a good probability Russia already knows everything he knows.