Russia wins.
They (as the USSR) win the first milestones hands down. Namely, the first successful craft to orbit, the first to lift an animal into orbit, the first to place a satellite into orbit and the first to put a human into orbit, and bring them back.
They achieved a bunch of other "firsts" as well, while we Americans tout the supposed "deal breaker": landing men on the moon. Sure, it's big and commendable, but it isn't the "ollie ollie oxenfree" that we've grown up hearing all the time.
According to many accounts touted by the US propaganda machine and unwittingly communicated by mass media, including school curriculum, the generally accepted assertion seems to be that once the moon shot had been achieved, the space race was essentially over.
Well you can't call something null and void just 'cause you see you can't win. That's called being a quitter. And by lying as much as possible about your loss, you're a sore loser.
The fact is, the US people didn't lose interest in space exploration in the mid-70s, but the war in Vietnam and greedy oil interests sucked up all the funding from the US government.
The Space Shuttle program had squeaked through but only because much of it had already been pre-funded and it was cheaper than some of the other potential paths we could have trod down. But the shuttle was nothing more than a work horse, really. It didn't advance any incredible space discoveries in its own right and we saw where some of those budget-minded cost cuts really hit hard. Just ask the families and friends of the crew members of Challenger and Columbia.
So now a year after the last manned space mission launched by the United States, we Americans are forced to watch our few remaining astronauts take off at 3 am our time from a far away launch site. Where? A Russian leased space center in Kazakhstan.
You gotta be in it to win it, folks.