As before, the answer to the first part is very simple. Money.
If you look at the costs of Mercury through Apollo 11 (1958-1969), the space program cost the USA around 34.8 billion dollars (1960's money). Adjust that 40 years forward with inflation and you're talking about around 221 billion dollars.
How many countries have that kind of coin to spend? The US did, the Soviets
thought they did (turns out they didn't - helped lead to some of the major economic problems that caused the fall of the Soviet Union). Japan? Nope. The EU? Nope. Central/South America/Africa? No way. China? Not until recently, and they're working on it. India? See China.
The other part of the equation is that since the Saturn V was (tragically) axed in 1971, the world hasn't had a heavy-lift vehicle sufficient to get a large payload to the Moon.
-The Saturn V could put 100,000 lb. to the Moon, 262,000 lb. to Low Earth Orbit (over half of that obviously being the propellant to get to the Moon). -The Space Shuttle can put around 53,000 lb. to Low Earth Orbit, most of that being the Shuttle itself.
-The biggest contractor rockets - Delta IV Heavy (ULA/Boeing): around 56,000 lbs. to LEO. Atlas V Heavy (ULA/Lockheed): 65,000 lbs. to LEO. -Internationally, Ariane 5 (ESA): 46,000 lbs.
As for how the US beat the Soviets... turns out the Soviet space program got very political, at one point with about 30 rocket designs competing for the same money, as opposed to the US, who picked one idea and pumped money into it. The Soviets actually were still ahead up until 1967 (Apollo 8). Another big problem was that the Soviet chief engineer, Sergey Korolyov, died (possibly of cancer) in 1965. Two years later, politics forced the USSR to launch
Soyuz 1 when it still had design faults, resulting in the death of a cosmonaut (They wanted a man on the Moon in '67, in time for the 50th anniversary of the October Revolution). The resulting year-and-a-half delay gave the US enough time to sprint to the finish line.
*edit* There's actually a great op-ed in the NY Times today about it:
http://www.nytimes.com/2009/07/18/opinion/18iht-edbignami.html