Although they became allies during World War II out of the necessity of beating the Nazis, the United States had a healthy distrust of the Communist regime, which took over Russia (renamed the Soviet Union in 1922) in 1917. After the war, it didn't take long for the US to clash with the Soviets in Germany and other places on the globe. This conflict never broke out into actual violence between armies (although clashes occurred between proxies of the two sides) and so was labeled the Cold War.
For all of its existence, the US has been considered a constitutional democracy with deep roots in the democratic ideals espoused by Enlightenment philosophers in England and France. In fact, most historians consider the US to be the greatest example of democracy in world history. One of America's greatest achievements is arguably the peaceful transfer of power every four years which occurs after elections. Since the middle of the 1800s, the US has been ruled by a two-party system with representatives of the Democratic and Republican parties dominating elections since 1860 (other parties have come and gone but have never been able to elect a president or take control of Congress). While each party has developed differing views on a number issues, they both agree in the rules set up in the Constitution. To this day, Americans still have free elections (despite recent attempts by Russia to interfere) and are able to choose their own leaders.
On the other hand, the Soviet Union was created by a group of revolutionaries who wanted to bring down the old institutions, mainly the power of the Czar, and establish a new type of government based on the theories of Karl Marx. Marx believed a new paradigm would occur when the working class (proletariat) rose against the dominant ruling class (bourgeoisie), overthrowing them and creating a people's republic. Unfortunately, Lenin and the other members of the Bolshevik party never quite got around to giving up power. Thus, not unlike its Czarist past, but without the familial transfer of power, the Soviet Union became a dictatorship, especially after the rise of Stalin.
Decisions in the Soviet Union, then, were made from the top down and often at the whims of one man. It's estimated that Stalin was responsible for the deaths of millions of Russians before World War II, during and after. Historians argue that because of this top-down style of governing the Soviets almost lost World War II with the Nazis stopped only miles from Moscow in the winter of 1941. Stalin and his successors continued to rule what would become a far-ranging empire after the war with the same style.
Economically, the US has, since its inception, believed in the principles developed by Enlightenment thinkers such as Adam Smith, whose treatise Wealth of Nations has been considered the bible of free-market proponents since its publication in 1776. Smith is often considered to be the father of capitalism. In the simplest terms, capitalism is an economic system which gives private ownership to the means of production. Private companies owned by Americans produce goods which they then sell on the free market. This system thrives on competition and the ability for individuals to market and sell goods at lower prices than others producing the same items. Over its history, and especially in the last 100 years, several American companies have risen to become immensely successful and wealthy, so much so that the American economy is by far the largest in the world and continues to grow.
In contrast, the Soviet Union, misinterpreting the theories of Marx, centralized the means of production with stringent government regulation and oversight. For example, during the reign of Stalin, family farms owned by Russian citizens were seized and collectivized. Not surprisingly, this created food shortages as well as bad feelings among the Russian peasantry. The Soviets struggled with agricultural production well into the 20th century because of collectivization. After World War II, unlike the booming US economy, the Soviets suffered through dour economic times for many years. Finally, with the arms race instigated by American President Ronald Reagan and the disastrous ten-year war in Afghanistan, the Russian economy basically collapsed in 1989 leading to serious reforms and the ultimate disbanding of the Soviet Union with nations such as Ukraine, Latvia, Estonia, Belarus, and Lithuania proclaiming their freedom.
Today, the US continues to be the greatest economic engine in the world, although China is now a close second. It also remains democratic with a two-party system. Russia, on the other hand, has only the twelfth largest economy in the world, just behind South Korea and Canada. The Russians have opened their country to capitalist investment but are mired in a dictatorship with the same leader, Vladimir Putin, in charge for almost twenty years.