Russia, and then the Soviet Union, had always lagged behind the rest of western Europe throughout the Industrial Revolution. Some of the more foresighted Tsars sought to overcome this. By the time the Communists had consolidated power they knew they needed to catch up economically and industrially and so they instituted 5-year plans. These were based more on building their economy and creating an industrial society comparable to the Wests rather than specific technological parity, although that was implicit in the plan.

Russian science was good, very good, but during this time they lacked the fiscal and industrial resources to match the West.

During WW2 the Soviets succesfully infiltrated the Manhattan Project in multiple areas and stole a vast amount of information from the Allies. The Soviet nuclear progrman then managed to 'solve' many of the problems of Atomic power by using this stolen information. This undoubtedly gave them the nuke long before they may otherwise have had it. This success is probably the most important one as it dissuaded the US from using its Nuclear arsenal as leverage because the Soviets were never far behind (and they often overestimated their capability as well).

Source: Richard Rhodes: The making of the Atomic Bomb.

There were other succesful spying attempts, although much of the Soviets efforts were based on the political situation. Most of the spying was obviously in the military technology area. Proximity Fuses are one that I recall, circa WW2.

The cause is obvious, firstly Russia/Soviet Union had long been behind the West, starting originally with Britain and then eventually the USA. Its a fairly obvious ploy to steal their ideas and use them yourselves. They had some top grade scientists, but just lacked the industrial power to be a leader, and their academic structures had to struggle through poor economic conditions and massive social upheavals. The Soviets attitude to scientists didnt help, they were a bit suspicious of such bourgeois activities such as academia.

The West were VERY interested in Soviet science, but primarily out of concern that they would pass their own in some field. This proved more of a false fear than any reality. They would certainly have snooped (altho much of that is probably still secret) but moreso to compare technologies than to actually learn stuff that our side didnt have.

When Viktor Belenko defected to Japan in his MIG25 in 1976 the craft was descended on by the US, tested, dismantled and then eventually shipped back to the USSR in boxes. I doubt they learned anything that was then adopted as it proved cruder than anticipated, but they still had a need to know. The soviet threat was often overestimated in the West and there was a constant fear that 'they' would get the drop on 'us' in some way. Popular fiction like Firefox by Craig Thomas (book and movie with Clint Eastwood) are an expression of this fear of technological dominance persistent at the time. Now looking back it seems exaggerated as the Soviets never or rarely matched technologies with the West.

No doubt there are exceptions that statement, but its doubtful that overall they ever had any clear advantage. But for example in the area of Tank Armour the Soviets were the first to make heavy use of composite armour, something ubiquitous in modern designs. They also were experts in reactive armour, more so than the west. Again however its doubtful how much of this made overall. In the area of computers and electronics they were far behind.