I've been fortunate enough to recently visit both Houston Space Center in Texas, and Memorial Museum of Cosmonautics in Moscow, Russia. These places have similar exhibits for tourists, but they are also different in surprising ways. Let's see how they fare in a different kind of space race.
Like many buildings in Moscow, The Museum of Cosmonautics is quite impressive at first sight: visitors enter through a park, and the museum has a gigantic monument depicting a rocket taking off on top. Houston Space Center entrance next to their parking lot is not quite as imaginative. Walking in from the Starbucks across the road was a very Texan pedestrian experience, that is to say I felt like I should have had a car. They do have a space shuttle perched on top of a 747 at the front, but even that doesn't compare to the sheer Soviet grandness of the Russian monument.
Best first impressions: Музей космонавтики.
Touchable space rocks
Houston Space Center gives its visitors a rare opportunity to touch an actual piece of moon rock brought down to earth by the Apollo astronauts. Or rather touch the layer of greasy substance that's formed on it throughout the years. Still, there's something memorable about the moment when one reaches for it through the secure display case. The meteorites casually left on a table in the Moscow museum are in better shape and interesting hunks of metal in their own right, but there's not as much magic in touching them.
Best touchable space rocks: Houston Space Center.
In Moscow, there are many informative 5-meter tall scale models of rockets, including a few with cutouts so you can see the engineering on the inside. They are nice to look at and informative, but don't come anywhere near to what the Houston Space Center has in store. In Texas, they had some leftover rocket hardware from the space program lying around, and they decided to make the best of it: visitors get to see the most powerful rocket to ever go to space in actual size. When first entering the large hall housing the Saturn V moon rocket, the view is awe-inspiring. It's curious to see the first stage of the rocket, a machine the size of a small apartment building, that was designed to be used only once for 168 glorious seconds of upward thrust.
Best rockets: Houston Space Center.
Footnote: When in Moscow, one does also have the opportunity to see some life-size spaceship models, but they're located at the nearby All-Russian Exhibition Center, so they don't count towards the score of the museum. Even if they did, they're hardly a match for the Saturn V.
Nods to the achievements of the other side
Both museums are clearly focused on the achievements of their country's space program. Museum of Cosmonautics reinforces Gagarin's status as a national hero, and even hails the abandoned Buran Soviet shuttle program as a success, while Houston celebrates the marvel of the moon landings. Still, international cooperation aimed at building peace during the cold war and on the modern ISS is also featured on both sides. Even as the situation between east and west has grown more tense during recent times, space cooperation is still going on. Splitting the ISS in two would hardly serve anyone's interests.
Best nods to the achievements of the other side: tied.
The American museum features a large mural painted by an astronaut celebrating US achievements in space and the glory of exploration. It's nice, but not that remarkable. The Russians on the other hand can't keep their cultured nature in check even in a museum of engineering, nor should they. There's a whole gallery of portraits of Gagarin in different styles, and also artwork featuring space visionary Tsiolkovsky. It's a great break from looking at a large collection of purely utilitarian objects.
Best artwork: Музей космонавтики.
The Space Center cafeteria is food-court style, with a selection of fast food options. It's large and comfortable, and the ceasar salad they offered as a healthy option was nice enough. The cosmonaut cafeteria is the exact opposite: a small side room that attracts visitors with foods that are packed cosmonaut-style, though they also have a menu of items to order. I can't say that the borscht soup in a tube would be something I'd want to consume regularly, but it's an experience.
Best cafeteria: tied.
If the Americans are good at something, it's selling things. There are fridge magnets, mission patches, clothing in all sizes, you name it. The cosmonauts' shop is much more modest in comparison, with a slightly odd selection of memorabilia including some Chinese Lego knock-offs.
Best souvenir shop: Houston Space Center.
Looks like the total score is 3-2 in favor of Houston Space Center. It's also operating on a higher budget and thus generally a bit more polished and modern. Still, I'd recommend going to both if the opportunity arises: each has its own unique merits, and together they present a more complete picture of the early history of space exploration. As an English-speaker, be sure to grab the audio guide in The Museum of Cosmonautics, as otherwise you'll miss some things that are only written in Russian.