The Soviet Union believed in spreading Communism (an ideology) through giving communist revolutionaries weapons such as guns, tanks, and planes.
The USA believed in projecting military power to guard its economic interests, which included containing the spread of communism (domino theory). Sometimes the US even gave weapons to bad people (Saddam Hussein, the Taliban of Afghanistan, etc.) to counter the spread of communism or guard its economic interests. The US has good reason to believe that it cannon rely on many of its fickle "allies." Therefore the US needs another method of projecting military power. That is where the carriers come in. They are much more reliable than dictators etc. However carriers are also much more vulnerable.
Nuclear Powered Icebreakers were more important to the Soviet Union (and now Russia) because no other ships (neither aircraft carriers nor tankers) can survive the ice of the Artic Ocean, which is where much of Russia's oil resources are located.
The Soviet Union did build aircraft carriers, but on a much more limited scale than the USA.
Communism prevented the Soviet Union from trading as much as the US, since it believed in centralized economies. Centralized planning also allowed convoys of ships to have military escorts such as those to Cuba. With the end of the Cold War, Russia has little need for naval power and Ukraine (former USSR) recently sold a partially completed Soviet aircraft carrier to China.
It should also be noted that the the most humiliating defeat in America's recent history was the attack on Pearl Harbor, which was conducted from Japanese aircraft carriers.