Two things are important to realize here.

First, the USSR was a federation of several bureaucratic entities (republics, oblasts, autonomous zones, etc.). The largest and most important of these entities were the Soviet Socialist Republics (SSRs). The Republics were largely determined by how Moscow defined and delimited ethnicity. The logic of this arrangement was that the Soviet Union's many ethnic groups would have their own "homeland" within the USSR. The largest SSR was the Russian Soviet Federated Soviet Republic/RSFSR. The majority of the population of the RSFSR were Great Russians (this is one of the shorthand terms used to refer to the Russian ethnicity and not confuse it with the Russian state or geographical entity- the distinction is present and more clear in the Russian language).

This segues into the second point, namely that the USSR was an ethnically diverse polity. Great Russians constituted approximately between 40% to 50% of the total population of the USSR. Even though the RSFSR was the homeland so to speak of the Great Russian people, there were still sizable numbers of non-Great Russians residing there. The Soviets enacted various policies that often operated against the interests of the Great Russian population. In drawing up the internal borders of the state, the Soviets territorialized ethnicity which created internal republics and other administrative entities to be led by the respective dominant ethnicity (e.g. Ukrainian, Kazakhs, etc.). Inside these new republics, the state facilitated indigenous language instruction, sponsored cultural activities, and, especially in the early years of the USSR, disciplined instances of Great Russian chauvinism. One of the animating policies of the USSR was korenizatsiia (indiginization) which sought to promote native cadres loyal to the Communist Party and the Soviet state. Although the Stalin turn of the 1930s pushed back against korenizatsiia, this idea never really went away throughout the life of the USSR. Entities like the Communist Party, the Komsomol, and the Red Army acted as supranational entities throughout the USSR and were quite prominent social institutions. The planned economy often redistributed industrial and other economic resources in ways that did not benefit the Great Russian core of the RSFSR.

So while Great Russians soon emerged as the "elder brother" of the Soviet family of nations, it was far from a position of hegemony. While the upper echelons of the Soviet state tended to be staffed by Great Russians, there were sizable numbers of non-Russians operating within the Soviet system and promoted by it. So while the public face of the USSR towards the West was largely Great Russian, the reality was more complicated.

This is why the phrase "Soviet Russia" was so common during the Cold War. It was often taken as a given that the USSR was another iteration of the Russian Empire and Great Russian chauvinism. old War-era commentators and historians, who were often highly partisan, frequently described the Soviet Union as an empire as part of a broader assault on the legitimacy of the Soviet experiment. For example, Richard Pipes would label the USSR as a "prison of the peoples," and draw connections to crackdowns in Eastern Europe with discriminatory policies against Central Asians and the Ukrainian famine. For Pipes and his ilk, the Soviet Union's anti-imperial ideology was merely a stalking horse for Great Russian hegemony.

But historians and other researchers have far more data on the actual experience of non-Great Russians in the USSR and how the USSR tackled its ethnic question since 1991. "Soviet Russia" does not really fit to explain actual Soviet history as it sidelines these issues for a somewhat lazy, Cold War-era formulation of the USSR.

Although there are exceptions, if someone uses "Soviet Russia" in 2018, it is an automatic red flag that this person usually does not know what they are talking about.