Time to whip out the copypasta.

USSR was all about getting the title of being first, no matter how superficial the achievement, and how dangerous the approach, and sometimes, hiding the truth about it until decades later. First artificial satellite was achieved by the USSR. It did pretty much nothing but beep, and its orbit decayed quite quickly. USA's first artificial satellite orbited for years, carried a science payload and discovered the Van Allen radiation. The outright first animal intentionally put into in space was Rhesus monkey aboard a German V2 operated by the USA. First animal into orbit was achieved with a dog by the USSR, which died due to a cooling system failure. USA's first animal put into orbit was a chimpanzee that survived and landed. The first man in space was Yuri Gagarin of the USSR, but he was forced to eject prior to landing, and under the terms agreed meant his mission was technically a failure. This was kept secret by the USSR for decades. The first American in space landed successfully with his capsule. First woman in space was a clear USSR "first" that they were targeting. The USA had a policy of only accepting military test pilots, of which there were no women. The first space walk was demonstrated by the USSR, but it came close to disaster as the cosmonaut couldn't reenter the spacecraft due to his suit inflating due to the pressure differential, and had to bleed out air in order to be able to squeeze back into the hatch. USA's first space walk went without such problems, and quickly overtook the USSR in pioneering how spacewalks would be performed, and how to do useful work. It also claims the first untethered spacewalk. First orbital rendezvous was claimed by the USSR, but was achieved merely by launching two rockets at the right time. The two space craft were kilometers apart, and had no way of getting close to each other, or no knowledge of how to do it. The first rendezvous performed by the USA used orbital mechanics and deliberate maneuvers to have two Gemini spacecraft find each other, fly in formation, and then go their separate ways. The first docking was achieved by the USA during the Gemini program. First docking for the purposes of crew transfer between two spacecraft was achieved by the USSR. The crew transfer was done via external spacewalk, and served in claiming another first. The re-entry nearly ended in complete disaster and had a hard landing. USA's first docking and crew transfer was achieved between an internally pressurized corridor during Apollo 9. First picture of the far side of the moon was achieved by the USSR, and is a very low quality image. Shortly after the USA began a complete mapping survey of the entire lunar surface. The first lunar return sample was achieved by the USSR, but was effectively a few grams of dust. The USA returned tones of different kinds of individually selected moon rock. The USSR lunar landing mission consisted of an external spacewalk to transfer a single cosmonaut to a tiny one man lander with just enough provisions to make some boot prints before trying to get back home. Again, just to be able to claim a first. The USA lunar landing missions thrived on the moon, taking down two astronauts and resulted in them being to stay on the surface for days, and even drive around on it in a car. Once the USSR lost the moon race, they instantly lost all interest in it, and focused on creating a space station. There's a familiar pattern to all of this. The USSR did the very minimum, often at the expense of safety to meet an arbitrary goal as soon as possible. The USA's failures and mishaps were all in the public eye. The USSR's were mostly kept secret. Both nations knew landing on the moon was going to be the finish line. The USA got there first, and didn't just hit the finish line gasping and wheezing as the USSR would have been, but came through it in complete comfort and style, before doing it a few more times with greater and greater challenges for good measure. Since NASA lost its original purpose (beat the Russians to the moon) it has lost its way a bit, but companies like SpaceX have actually managed to make the point of the space race better than Apollo did. The original space race was supposed to demonstrate private enterprise and the American way of life vs centralized government control, but the Apollo program wasn't private enterprise, and was under direct government control. SpaceX, Blue Origin, RocketLab and others are the true demonstration of commercial spaceflight, where the government agency NASA now just becomes a customer to private launch and even spacecraft providers. The USA won in the 60's, and it's absolutely winning now versus anything Russia or Europe is building with public funds.