Speaking at the museum, Mr Armstrong said the Moon race was a "diversion" in the Cold War battle between the US and the USSR.
"Eventually, it provided a mechanism for engendering co-operation between former adversaries. In that sense, among others, it was an exceptional national investment for both sides."
Fellow astronaut Dr Aldrin praised President John F Kennedy's bold decision to pronounce that the US would land men on the Moon before the end of the 1960s.
"Apollo 11 is a symbol of what a great nation and a great people can do if we work hard, work together and have strong leaders with vision and determination," he said.
But he also pushed for a mission to Mars: "The best way to honour and remember all those who were part of the Apollo programme is to follow in our footsteps; to boldly go again on a new mission of exploration."
Mr Collins, who circled the Moon alone while Mr Armstrong and Dr Aldrin walked on it, said Mars was more interesting than the Moon.
"Sometimes I think I flew to the wrong place. Mars was always my favourite as a kid and it still is today."
My glass has been half empty for three decades at least. Hopefully, we can turn that around because what we did then is do-able again
Eugene Cernan, former Nasa astronaut
He urged further exploration, saying Mars was a "much more worthwhile destination".
Other Nasa astronauts gave a news conference at Nasa headquarters in Washington DC on Monday.
Eugene Cernan, who was the last astronaut to step off the Moon, in 1972, concurred with the Apollo 11 astronauts and called for a new focus on Mars.
"We need to go back to the Moon, we need to learn a little bit more about what we think we know already, we need to establish bases, put new telescopes there, get prepared to go to Mars. The ultimate goal, truly, is to go to Mars," he told journalists.
"I think the next major goal is not to spend three days, or three weeks or three months on the Moon, but to have you folks, or your kids, or your grand-kids sit here and talk to a group of guys who can tell you what it was like to go to Mars."
Neil Armstrong: 'The ultimate peaceful competition'
But Dr Aldrin disagreed with the view that astronauts should test capabilities for a long-duration flight on the Moon before attempting a journey to Mars.
"Why go to the most difficult place to do that? Why not do it on the International Space Station," he said.
Dr Aldrin added: "One day, we are going to send some people to the surface of Mars. And if we think we're going to send them there for a year-and-a-half and then bring them back, and then send another group there for a year-and-a-half and bring them back, Washington will find another way to spend that money.
"That's unless we have declared our objective is an increasing, permanent space settlement."
The US space agency's currently stated aim is to return astronauts to the Moon by 2020. But that vision is under review, along with the space vehicles that would get them there.
Nasa is due to retire its space shuttles next year and replace them with the Orion spacecraft, an Apollo-like capsule that would launch on a new rocket called Ares 1.
Another rocket, Ares V, would have the capability to launch heavy payloads - service and cargo modules - that would be needed to service Moon missions.
What should Nasa tell its astronauts to do on Mars?
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.