What are the benefits of warming up and cooling down?

We suggest that people warm up for a minimum of 12 minutes (I do 15).

Warming up is not specifically about getting the heart rate up there, although it is a part of it. When the body is at rest, the majority of the blood is moving around in the internal organs, and relatively little of it is moving around in the muscles. The function of the warm-up is to slowly increase activity levels so that the blood has a while to leave the organs and head over to the muscles. (A slow increase in heart rate helps drive this process). But if activity has a very quick onset, the body is going to get the message that the leg muscles need blood immediately. This means that the muscles in the core are going to contract, the heart rate is going to skyrocket, and the valves within the blood vessels are going to open and close such that blood is shunted to the legs.
The combination of the stress (which is what allows these changes to happen quickly) and the rush of blood exiting the organs can cause the organs to go into shock. ( Shock is defined as "the abrupt and undue interruption of a process"). Even if the organs don't go into shock, the more abrupt the changes, the more they wear down. In other words, the reasons you want to warm up the body are the same ones why you want to warm up a 1980s-model vehicle.
The cool-down is as equally important, but for other reasons. When you don't cool down, and your heart rate immediately drops, the blood currently within the muscles doesn't get a chance to be drained out of them. Consequently, blood pools within the muscles. This is "dirty" blood, containing relatively large quantities of CO2, acidic hydrogen ions, and other by-products of exercise. The longer it stays within the muscles, the more of a chance it gets to damage them. And because the muscles are still being damaged, they are not going to start recovering in earnest until that blood has had a chance to leave.
So, the warm-up allows exercise to be a less stressful endeavor for the body. Cool-down allows recovery to happen effectively and in a timely fashion. (12 minutes is the minimum time it takes for each of these processes to run their course).