I think we need to question some of the assumptions here.

First, a country's total GDP (the basic equation for GDP is Personal Consumption + Investment + Government Spending + Net Imports/Exports) doesn't necessarily indicate how "rich" a country is, at least in the way we would commonly think. GDP (or its close relative, GNP) is basically measuring how productive a country's economy is on a countrywide basis. This method of measuring an economy's total size will see increases if productivity increases, or if the raw population increases. So countries with big populations will almost always be near the top for GDP totals, and in 1970 the USSR had the third-largest population of any country in the world, with about 241 million people (behind China and India, and ahead of the United States' 209 million). A useful modern day comparison would be how China's GDP (depending on how you count it, either at nominal exchange rates or Purchasing Power Parity) is either first or second place in the world, despite the GDP per capita in China being about $8,600 in nominal rates, versus almost $60,000 in the US.

And this gets to the second issue. Generally, when we say a country is "rich", we're talking about GDP per capita, rather than total GDP. A country can have a relatively high total GDP if it has enough people to offset a relatively low GDP per capita. The estimates of Soviet GDP (more on this in a minute) placed it at around a third of US GDP in 1970, but that is with a total population that was 15% larger than the US population, meaning that Soviet GDP per capita was significantly below that of the United States. And even before the Soviet economic decline in the late 1980s, the annual Soviet GDP growth was in ever-smaller percentages (and behind that of the United States).

As for why the difference between the relative size of the Soviet economy and that of Russia - first, remember that Russia is not by any stretch of the imagination the same as the Soviet Union. Even in the 1989 census, the population of the Russian Soviet Federative Socialist Republic was barely around half of the total Soviet population of 292 million. Russia also went through severe economic and even demographic decline over the 1990s as it transitioned from a centrally planned economy to a (for lack of a better description) state-managed capitalist economy. The population of Russia has stabilized but is smaller than it was in 1992. The size of the economy decreased by an estimated 40% over the 1990s, and only stabilized at the turn of the century, and didn't regain its total GDP size until the first decade of the 21st.

And this is over a period of some twenty years from 1987 through 2008 which in US economic history is known as the "Great Moderation", where recessions were generally short and relatively shallow, and there were massive growths in productivity associated with the adoption of computer and internet/communications technology. In short, the Russian economy went into severe decline and played catch-up with itself for the first decade and a half or so after the Soviet collapse, while the US economy largely continued to expand. Note: even without productivity increases, the US population increased by over 100 million between 1970 and today, so that alone would massively increase the total size of US GDP.

And finally the note on Soviet GDP: almost any number you read for Soviet GDP is an outside estimate, and these estimates could differ wildly. The essential idea of GDP is that it measures additions or subtractions in value of goods and services produced in an economy, but in the Soviet case where the economy was centrally planned and almost completely state controlled, prices simply didn't function in the way they do in a market economy. The values for goods, especially things like capital goods, were basically accounting estimations that had both flawed and largely untranslatable assumptions built into their figures (and so were far less reliable than, say, comparing raw output figures like tonnes of steel produced annually, and even those figures were subject to massaging). So the USSR in its official statistics might say it was the second largest economy in the world, but most of those values were, say, because that's what the Soviet planning commissions considered the value of it's dams, or apartment buildings, or power plants to be - none of the costs in building, maintaining or improving those things were being paid for on a market, but by state allocation of resources. But Western economists and intelligence analysts would try to "adjust" those figures to be comparable in a meaningful way with another country's GDP, and sometimes these adjustments could produce dramatically different results from the official statistics, and even from each other.

Sources

Yuri Dikhanov. "A Critique of CIA Estimates of Soviet Performance from the Gerschenkron Perspective" in International and Interarea Comparisons of Income, Output and Prices talks about the history of methodological differences in measuring the size of Soviet GDP.

Stephen Kotkin, Armageddon Averted

Alec Nove, An Economic History of the Soviet Union.