NRO's Corner is discussing if the Apollo program helped end the Cold War and bring down the Soviet Union. They post the thoughts of a Soviet military officer as part of the discussion.
I believe that the Moon landing was the most consequential American action since at least WWII and perhaps since the 18th century. There was definitely a huge impact on the Cold War. There was a tremendous propaganda effect (and let's not forget that in the absence of American landings the Soviets would sooner or later have landed on the Moon themselves and made a huge deal out of it).