Why did ideological differences cause tension between the United States and the Soviet Union during the Cold War?
The specific ideological differences between the US and USSR, or why did their differences cause the conflict?
The differences are pretty obvious, the US being a country based on the principles enumerated in the Declaration of Independence, the Constitution, Thomas Paine's books, etc. No matter how we've fallen short of those ideals at times, we have a society based primarily on the ideas of personal freedom and personal responsibility. Our government and persons of influence have, at times, exploited the population and our laws for their own power and gain, but the ideals of our society usually keep the damage from being too great to be corrected by reforms.
The Soviet Union was also based on an ideology, the Marxist concept of Dialectic Materialism, a sort of theory of history based on the tensions of ideals (thesis), their opposite (antithesis), with the tension resolved in eventual synthesis. Marx created an entire theory of science, historical analysis and economics based on this. Unfortunately, he knew very little about science and economics, and his view of history was extremely one-sided. The practical problem of the Soviet Union was not the theory, but the fact that those who seized power in 1917 wanted that power for its own sake. The Soviet government wanted to rule all that had been Tsarist Russia and to continue the expansion of their power through the use of blunt force, ideological propaganda, and terror through the secret police and the threat of punishment (prison, work camps, torture, and death).
The chief difference is this- the ideology of America encourages independent thought and personal responsibility, that of the Soviet Union sought total control over the thoughts of the citizens, either by literal mind control through propaganda or the threat of punishment if one spoke or acted along independent lines.
As to why this caused conflict, when the Second World War ended every inch of Europe the USSR had liberated from the Nazis became either part of the USSR (the Baltic States), or the Soviets put in power governments they controlled, and then used their military power to ensure they could not be driven out without another world war. So you have the United States espousing an ideology completely opposite that of the USSR, both with huge armies facing one another across the corpse of Germany, neither one really wanting total confrontation, but neither willing to completely back down.
Given the Dialectical view that the whole world must be driven down the path of revolution, conflict between the Soviet government and the entire Western World was inevitable, the entire structure of Marxism being inimical to the traditions of Western philosophies of freedom.
During the cold war era, tensions were hot and heavy between the US and Russia because of the differences in governmental ideology. The United States, a democratic republic, was almost diametrically opposite of the USSR, that is, the United Soviet Socialist Republic.
The communist regime that had started with Lenin and Marx was still alive and flourishing inside the borders and boundaries of mother Russia, as its red flag with yellow hammer and sickle made a powerful statement of the "worker's paradise" mentality still held inside that nation. The United States, meanwhile, was in the middle of the Reagan era for a good portion of the cold war, and it was flourishing as a result. Reagan's trickle-down economics meant success not only for the rich, but for those who were under them. As America "went big" in every way, Russia continued to wallow in the archaic and broken system of communism, and animosity was the result.