Could it be the USA and USSR had different post-war aims, the USA and USSR both broke promises which contributed to the breakdown in their relationship? Or could it be that the USA and USSR had major ideological differences the USA and USSR believed there could only be one superpower in the world?
The USA (democracy) and the USSR (communist) were diametrically opposite of each other. They had very little in common. Very different ideas how the economy should work.
Their alliance was one of necessity. They needed each other to defeat the Axis powers. Once that was achieved, it was a matter of time before they became political enemies.
The Americans send an expeditionary force to Siberia in 1918. They withdrew that force in 1920. Mainly because the communists had won the civil war. The relations between the two were not 'the best of friends' to say the least in the 30's.
Ideology was secondary, spheres of influence were primary reason for geostrategic rivalry
Before WW2 both US and Soviet Union were more of regional then global powers. US had great influence in North and South America (Monroe Doctrine), but almost no political influence in Europe, and only limited influence in Asian Pacific region due to other powers like Great Britain and Japan . Soviet Union as only socialist country had some global influence trough various leftist movements, but this was limited. In Europe, other powers (Great Britain, Germany, France etc ...) effectively blocked Soviet Union, and only after the start of WW2 USSR managed to act locally in Finland and Romania (Bukovina). In Asia USSR clashed with Japan over Manchuria and Mongolia, and had some influence in Central Asia . Although one across another in Bering Strait, USSR and US didn't have much contact and were not seen as opponents despite different ideologies before WW2.
After WW2 things changed dramatically. Both Germany and Japan disappeared as great powers. France was severely weakened, and so was Great Britain (British empire collapsed soon after) . Although badly hurt in the war, Soviet Union was now undisputed victor in eastern part of Europe, and ideologically aligned itself with numerous leftist liberation movements in Asia (most important of these was in China) . US became leading power in the West and effectively succeed Great Britain. In fact, after WW2 Britain was practically unable to have independent foreign policy (as witnessed for example in Suez Crisis) and become US junior partner. During the war, Churchill and Stalin agreed about zones of influence in post-war Europe, now British zones simply became US zones. In China, both US and USSR had their favorites, and Soviets won. There were also divisions in Korea and Vietnam which later led to wars.
Although ideology appears to be at the forefront of the conflict, in reality it was not so . Fr example, US (supposedly champion of democracy) often supported various monarchs and dictators in order to thwart Soviets. They were even ready to support communist China, after Sino-Soviet split. Soviet Union, although nominally more ideological, didn't shy from helping completely anti-communist regimes when it suited them. For example, in 1980 Argentina increased exports of food to USSR, despite other countries in Western hemisphere boycotting it because of Soviet intervention in Afghanistan. To return the favor, Soviets abstained in voting for Security Council resolution 502 and allegedly helped Argentinians in Falklands war with some recon assets.
As a conclusion, we could say that after WW2 both US and Soviet Union became global powers with conflicting interests in various parts of the world. As other countries joined one or other side conflict intensified and became Cold War . Ideology was important part of this all, but it was not deciding factor - after all, Russia is no longer socialist country yet it is again opposed to US although ideologically and culturally much closer then for example US ally Saudi Arabia. Rather then that, strategic interests decide who would be ally and who would be foe.
The reason was best described in 1835 by one Alexis de Tocqueville in "Democracy in America."
"There are at the present time two great nations in the world, which started from different points, but seem to tend towards the same end. I allude to the Russians and the Americans...
All other nations seem to have nearly reached their natural limits, and they have only to maintain their power; but these are still in the act of growth... The Anglo-American relies upon personal interest to accomplish his ends, and gives free scope to the unguided strength and common sense of the people; the Russian centres all the authority of society in a single arm. The principal instrument of the former is freedom; of the latter, servitude. Their starting-point is different, and their courses are not the same; yet each of them seems marked out by the will of Heaven to sway the destinies of half the globe.”
Exceptions to the above "rule" occur when the two have one or more common enemies. Up to 1945, these consisted of Germany, Italy, Japan, and a few east European nations. In the 1860s, Russia was still "smarting" from its defeat in the Crimean War by Britain, France, and others. It was then very friendly with the United States, who feared British and French recognition of the Confederacy (and later sold Alaska to the reunited U.S.to create a "buffer zone" between British Canada and Siberia)
In 1945, the "common enemies" disappeared, and America and Russia reverted to the "Tocquevillian" order. They will become friendly again if and when there is a common "third" enemy (e.g. China).