Each has its own purpose, both related to performance as well as injury prevention and safety.
The warm-up is to prepare the body for the demands of the exercise to come later - a rehearsal of sorts for the more intense movements to come. Just launching into full-scale activity without gradually warming the muscles (even for a few minutes) can increase the risk of injury dramatically.
In addition, the gradual build of the warm-up and the initial lead-in into your aerobic training intensity allows the body time to get the heart rate, breathing rate, etc., up to the levels you will need in order to sustain the higher level of activity. If you increase the intensity of the exercise too quickly, you can limit your performance - just as with going from resting to a sprint, lactate buildup in the blood (the byproducts of anaerobic metabolism) will cause you to need to slow down.
As for the cool-down, the gradual cessation of exercise helps promote venous return - the massaging action of the muscles helps with blood flow back to the heart. Stopping suddenly can cause a sudden drop in blood pressure, which can leave you light-headed or dizzy. In addition, for some people with latent (or diagnosed) heart disease, sudden cessation of exercise can bring on cardiac arrhythmias, which can be fatal.
--------"It's my personal opinion that people need to be more clear about what is fact and what is in fact their personal opinion." -- me