During the Cold War, the United States and Soviet Union were competing with one another for global and political dominance. With that in mind, neither side could afford to fall too far behind the other in its scientific and technological capabilities. These concerns, as they related to the power dynamics between these two States, were driving features which shaped both the Space Race and the Arms Race.

In some ways, the Arms Race is the more easily grasped of the two. Ultimately, the stakes here are fairly clear cut. The Arms Race refers to weapons technology (most famously to weapons of mass destruction), with both sides spending tremendous energy building and maintaining their own nuclear arsenals, as well as improving the technology involved.

As far as the Space Race is concerned, one should not forget the degree to which it was itself actually quite closely linked to the Arms Race. For example, consider that Sputnik was launched into space by the world's first Intercontinental Ballistic Missile. In that respect, the Space Race has its very origins in the Arms Race. Irregardless, space technology certainly had military applications, not only in terms of the militarization of space (there were concerns that the Soviet Union would establish nuclear arsenals in orbit) but also in terms of more general technological and scientific capabilities and knowledge.

In that respect, I would say that the Space Race and the Arms Race were closely intertwined. Regardless, the stakes were high, and in both cases, neither country could afford to ignore the progress (or potential progress) of its adversary, lest it risk falling behind on the technological curve. Scientific and technological advancement was one of the critical theaters in which the Cold War played out.

Both the United States and the Soviet Union feared each other's nuclear weapons, which is understandable.  They also feared a "missile gap" where the other superpower had more missiles or warheads, and so they frantically constructed more weapons to try and maintain an advantage strategically.  In addition to constructing a huge arsenal of atomic weapons (the total by 1985 between both countries was a staggering 27,000 nuclear weapons), both sides built and maintained huge standing armies in preparation for what they felt was an inevitable World War III.

The Space Race seems almost silly today, when you consider that Sputnik was only capable of beeping a signal back to Earth and nothing more.  But in a nuclear context, America feared that the Soviets could drop nuclear warheads from satellites without warning, so we felt it was absolutely necessary to "catch up" to the Russians in technology.

The arms race and the space race were related, but different.  The arms race was a competition to have the biggest and most dangerous weapons arsenal.  This was clearly important because each side was afraid of the other.  They each wanted to make sure the other could not overpower them.

The space race was a competition to get into space (first in space, first in orbit, first to the moon, etc).  This was related to the arms race because the rockets used for that could be used for nuclear missiles too.  It was also important because both countries wanted to look like they were stronger and more advanced technolgically (if they could be first in these various competitions in space).