I'll try not to open up too many cans of worms here....
So the Russians thought/knew it was a hoax - so since it was a "hoax" the Russians have less to worry about. If your rival claims he went to the moon, and you know he didn't, than obviously you have less to fear since they don't have the weapons/tech they claim.
If the U.S faked it, they were obviously trying to flex their "muscles" and show the Russians who's the boss. But the Russians KNEW it was fake so what did that accomplish exactly?
Here is what was accomplished. A couple of years before this supposed moon landing the Russians were the first to launch a satellite into space...it was called SPUTNICK. This accomplishment made america look quite IMPOTENT as american has always lauded itself as being so superior to every other nation. For the sake of not being out done, america had its president at the time: Kennedy, announce that it was going to go to the moon. Essentially one upping Russia on these space adventures. Russia soon sent up dogs, monkeys and then men into space all to americas chagrin. America then planned this fake moon launch and landing to try and keep its facade of superiority. Much in the form of specialized cameras, fake moon landscape effects, television and radio announcements made america and the whole world excited about this event of travel to the moon...which dwarfed Russias actual adventure into space.
Once america pulled off this hoax it sat back on its laurels never to try again returning to the moon or going any further into space to places such as Mars or beyond. As a matter of fact america has never returned to the moon nor will it allow investigators to see the ACTUAL mission capsule (they only show a model) neither will they allow anyone to investigate the supposed space suits or the infamous back packs that accomplished everything from providing oxygen, humidity, carbon dioxide cleansing, pressurization, air conditioning... of which no suit today regardless of cost is capable of doing despite the more advanced technology of this time...so how did they accomplish it then?
The Russians are still ahead of the americans as they are still the only people to perfect staying in space for extended periods of time, building space stations etc. America had to learn from teh Ruskies how to do that.
So when it is all said, it is all about dick measuring (I hate to have to say that), and keeping the facade of being greater.