The Cold War was a rivalry in which began after WWII between the two new "super powers:" USA and USSR. The Cold War became an arms race and eventually escalated to become way too out of control.
The Space Race was competition between Soviet Russia and the United States to achieve the realm of space travel.
The biggest influence the Space Race had to do with the Cold War was the development of the Intercontinental Ballistic Missile (ICBM). The creation of the ICBM was the biggest turning point on the Cold War, leading to arm races in which was the central to the Cold War.
Hope this helps :)