The main effect of the Apollo Project was not scientific, it was psychological. Sputnik had scared the Americans (and the rest of the world) with Soviet technological superiority. The response to this was a push for science and technology in the US, of which the Apollo Project was part Beating the Russians to the Moon restored the confidence of Americans (and the rest of the world) in US superiority.
The alternative to the push towards science would be a push towards religion. This happened, but would have even stronger. If US could not be the undisputed master of the material world, it could at least be the master of the spiritual world. New ideas and technology would no longer come from the US. US military would still be formidable, but without the crushing superiority. US would look inwards, only keeping enough of an active interest in world affairs to keep the oil flowing. US had saved the world from dictatorship in WWII, from now on, the world could look after itself.
Europe would be unable to fill the power vacuum, although they would be a leader in science. France and England would still be occupied with crumbling empires, and preventing West Germany to take leadership. They would fall under influence of the Soviet Empire. The Soviets were always numerically superior, but they were scared of US technology. With that gone, they would be even more expansionist. Their economic system would still be doomed eventually, but by expansion they would be to delay the collapse. Without the protection of the US, Japan would see the need to regain their military might, while keeping their economic growth. Japan would become the new leader in technology. Mao's China would suffer from the pressure from two sides (Soviet and Japan), and never develop into a new superpower.
So, at this time, the America's would still be under US influence, Africa, Europe, and South Asia would be under Soviet influence, and SE Asia (and Oceania) dominated by Japan. Technologically, we would be 20 years behind. The Soviet and Japanese empires would both start to show signs of recession, while the US might be rediscovering science which would eventually lead to a come back on the international scene.