What were some similarities between the Space Race and the Arms Race?

Both the Space Race and the Arms Race were part of the Cold War struggle between the United States and its market-driven, democratic ideology, and the Soviet Union, with its communist, totalitarian ideology. Both races were bound up in the struggle for technological supremacy that came to define post-World War Two foreign policy for the two superpowers. The Arms Race shifted the balance of power back and forth several times in the late 1950s and early 1960s, when the United States and Soviet Union were trying to build up their nuclear arsenals so that each could claim nuclear superiority, and maintain their so-called First Strike Capability, in the event of a nuclear war.

While the United States was ahead of the Soviet Union in enlarging its arsenal of nuclear weapons and miniaturizing warheads to place on top of Intercontinental Ballistic Missiles (ICBMs), the Soviet Union stunned the Americans in 1957 by launching Sputnik, the first man made satellite to enter Earth's orbit. This launch was deeply unsettling and embarrassing for the Eisenhower administration and for the United State's public relations, which aimed to convince unaligned countries that it was the better ally to have. 

For the United States, at least up until the launch of Sputnik, building up its nuclear capabilities and containing the spread of communism through proxy war, both overt and covert, was the main priority. Once the Soviets beat the United States into space, however, and showed that they weren't a second rate technological innovator, the Space Race became a main forum for the United States and the USSR to wage their ideological and public relations battles.

The fear on both sides that other country would colonize space first and weaponize it led to a massive outlay in spending from the pentagon, and when JFK announced his intention to land a man on the moon within ten years, the real competition began. The Space Race was not just a race into space, but a race between two diametrically opposed ideological forces who wanted to prove their supremacy to the world. Both sides understood that whoever got a manned mission into space first would claim bragging rights for eternity, and that kind of propaganda victory could win or lose the Cold War. Whoever won the space race would likely win struggle, and export its ideology and power around the globe. As it happened, the United States landed its astronauts on the Moon in 1969 and went on to win the Cold War, proving this thesis. 

Finally, the Space Race and the Arms Race were not separate races, but rather, they became conflated when the United States and Soviet Union realized that the military applications of space technology such as communication satellites could determine which side had more effective nuclear arsenals. Both sides used their space technology for missile guidance and detection of enemy missiles and enemy positions. The ability to place spy satellites in space and to use low orbiting ICBMs to launch nuclear attacks changed the face of warfare forever. Once the United States and Soviet Union realized that they could launch missiles with nuclear warheads from the other side of the world and have those missiles hit their targets in 35 minutes, instead of the  hours it took for bombers to deliver their payloads from nearby bases, the idea of having first strike capability became a moot point. Both sides used their space satellites to set up missile shields, that could detect incoming missiles and self-launch a so-called Massive Retaliation, which would ensure that a nuclear war, regardless of who started it, would end with the destruction of human civilization. 

This realization brought about the notion of Mutually Assured Destruction, and convinced both the United States and the Soviet Union that nuclear weapons had no practical use except as deterrents. In some ways, the Space Race actually brought the two countries closer together, making both of them realize that treaties such as Open Skies, SALT (The Strategic Arms Limitation Treaty) and a ban on weaponizing and colonizing outer space exclusively, was the best way to avoid a nuclear holocaust. The Space Race also brought about unprecedented cooperation between the two countries, who finally understood from the vantage point of outer space that they shared a planet, and needed to learn how not to destroy it, even as they both sought to defeat on another.