But we were not _really_ behind. We were about a year behind (if not less), and much was due to other American advantages: we didn't need huge rockets (which would launch humans into space) since American nuclear bombs were smaller.
The reason we don't hear much about China is because that don't do much:
Launch a man into space once a year? Land a rover on the Moon? That's 1960s news.
If it would be more serious, there would be a "Sputnik scare" like in the 60s, not silence
The big difference between the execution of the US and Soviet space races is that the Soviets produced the Soyuz - a very capable space craft and launch vehicle that still flies today. On top of that they pioneered space stations and long duration manned space missions.
The US on the other hand built an impractically large moonshot rocket that was too expensive to keep on producing. Then the US moved on the the Space Shuttle, which overpromised and underdelivered (e.g. did not serve the air force and get funding from there) and didn't provide a contingency into the future and had to be retired leaving the US with no manned space launch capability.
I didn't mention them (but I'm very well aware of them) because they're not very relevant to the point.
Skylab was a great project and good utilization of leftovers from the Apollo program (although the Soviets did much better with Salyut -> Mir -> ISS when it comes to technology reuse). The end of Skylab was a bit embarrassing as it did not survive long enough to be serviced by the shuttle and there were no more Saturn launchers remaining. Although the damage sustained at launch/deployment makes it arguable whether it would have been a good idea in the first place.
Hubble is also neat (and a testament to the capabilities of the Shuttle as it was repaired twice), and JWST is going to be a good successor.
But there's still very little continuity in the American space program compared to their Russian counterpart. I guess it's partly to blame on the fact the the NASA budget and goals change dramatically when the occupant of the White House changes but I've read some news that they're trying to establish longer term goals. I'm afraid that whoever is the next president will again move the goal posts and set back the SLS + Orion program, which is closest to being the first manned deep space capable space program after the Apollo program.
My point of comparison is the fact that variants of the Russian Soyuz rockets and space craft have been in continuous operation since 1966.
Sputnik was the science payload - at that point the US were still saying it'd be years until orbit was reached.
Ironically, had it not been for the space race, we might have a more mature space program now, rather than one largely based on 1940s technology and engineering. They rushed to production and massively invested in a single model on both sides, rather than exploring options fully before commencing - similar applies to nuclear technology, insofar as thorium would have been the better tech, but was too late to the party to gain traction, never mind the weapon byproduct bit.
The Western world isn't falling behind. It's just not so push-over ahead.
What China is doing right now, the USSR did in the 1960s, and the US did in the 70s (it had a ten year detour to go to the moon first).
While NASA hasn't done anything innovative in manned space-flight since 1980 (sunk costs, politics, no _real_ need, etc), they've done crazy work with unmanned spacecraft.
Think landing a _car_ on Mars (and then driving it around), launching routine missions to outer planets (and beyond!), being the only agency to get a probe out of the solar system.
Honestly, if Congress ever feels it necessary to launch someone to the Moon, they'll get it done in less than a decade.
And 'driving it around' is not as simple as it might sound, due to the time taken for commands to travel 4 light-minutes (minimum, and almost always a lot more). Feedback similarly.
The reason we don't hear much about China is because that don't do much:
Launch a man into space once a year? Land a rover on the Moon? That's 1960s news.
If it would be more serious, there would be a "Sputnik scare" like in the 60s, not silence