adventure asked me to elaborate on what I said before. I sent him a response privately but I have decided to share it with everyone else (and feel free to correct me or add to it).
How has the space race benefited mankind?
The Environment
Without the space program weather forecasting would not be as accurate as it is today. Satellites warn us before severe weather such as hurricanes and tornados hit us, reducing the amount of harm they can cause.
We are able to monitor the state of endangered species, the destruction of rain forests, the expansion of deserts, tsunami and earthquake damage, oil spills, volcanic eruptions, and more, all using satellites. They have made us more aware of the effects we have on our environment.
Communication
People take for granted how easy it is to communicate now. Major news events are broadcast around the world as they happen. We can communicate with other people in distant countries almost as easily as we can with our next-door neighbour. It brings us together and makes the world a smaller place.
Live broadcast television was improved greatly as a result of the need to transmit video of the Moon landings back to Earth.
Computing
Although computers were not invented for or by the space program, their development was certainly helped a great deal by it.
International Co-operation and Peace
Although the space race was a competition between two cold war foes, it was peaceful. By the middle of 1975 the Americans and Soviets were actually working together to make the first international space docking (the Apollo-Soyuz Test Project) possible. So, rather than heating up the hostility between the two nations, the space race actually helped create peace.
Knowledge
We understand our world, our solar system, and the entire universe, better as a result of our exploration of it. Whether it is through manned or robotic exploration, the knowledge we gain is priceless.
Medical Advances
There are many examples of medical advances that are a direct result of the space race. Pacemakers, biorhythm monitors, CAT scans, and even cold medicines are examples of how something created for NASA have worked their way into common use.
Safety
Smoke detectors, fire resistant fabrics, fire extinguishers, etc. were all either invented or improved by the space program.