This is heavily dependent on the era you are looking at. My MA thesis (in progress now) is looking at the Soviet nationality policy as it relates to one of their autonomous regions (Abkhazia) which itself was in one of the republics (Georgia). As I'm writing on the era 1917-1953, I'm going to discuss that here.

First, there was no Russian SSR, but rather the RSFSR (Russian Soviet Federative Socialist Republic). Thus while all the other republics had their own Communist Party, Academies of Science and Culture, universities, etc, Russians did not. This stems from the concept of Great Russian Chauvinism that Lenin feared (in short, it meant that because the Russians had the Empire, they were the most advanced people, and the non-Russians were discriminated against; Lenin wanted to end this and get the non-Russians up to the level of development of the Russians). So all non-Russian groups of a certain size were given their own territory, with their population determining the status (full Union republic, like Ukraine or Kazakhstan; autonomous republic, like Tatarstan; or autonomous region, like North Ossetia).

For the first several years of the USSR, a policy known as korenizatsiia (roughly, indigenization) took place in the non-Russian regions. This has been referred to as effectively "Affirmative Action" by Terry Martin, who is probably the biggest authority on the policy today. Much like affirmative action in the US, it promoted minorities at the expense of the majority; in the case of the Soviet Union, it thus meant that non-Russians were given positions within their own region and places within the Party apparatus at the expense of ethnic Russians. This was done for a couple reasons: one, to bring these people up to the same level of development as the Russians (as noted; during the Empire they were largely ignored); and two, to develop socialism because as Lenin and Stalin understood Marxist theory, nationalism was a by-product of capitalism, but once it was realised it could be moved on from and then socialism could be reached.

So places like Ukraine (I'll use Ukraine as an example as Martin's book focuses on them) saw a huge amount of Ukrainianisation policies. Whereas Tsarist Russia had banned the use of Ukrainian and all that, its use was encouraged, to the point that even ethnic Russians were forced to use it rather than their own language (if they fell into one of the regions that was considered Ukrainian; Russian regions were allowed to use Russian). The culture of Ukraine was promoted, and made to be distinct from Russian, to the point that even ethnic Ukrainians felt it was too much.

By the early 1930s it was apparent that korenizatsiia had not worked. Rather than foster any real development, all it did was increase nationalism amongst the non-Russians, and anger the ethnic Russians, who were not happy about being discriminated against officially. So this policy was slowly reversed, and Russification became the norm across the USSR. In 1938 it was made mandatory that every school in the USSR had to teach Russian (it was not required in the non-Russian regions until that point), and Russian culture was promoted as the culture of the Soviet people. Russians in effect became "first among equals" and were given prominent positions in the republics. With the Second World War this Russification was amplified, as it was a useful tool to unite the state against the Nazi invasion.

That should give some idea about the differences between the regions and peoples. In short, the 1920s saw ethnic Russians (not just in the Russian SFSR) discriminated against in favour of non-Russians, only to have this policy reversed when it was proving to not be effective.