Driving requires considerable brain power and the President needs to devote all his energy to solving the problems this nation and the world faces. Besides, what happens if he or she injures themselves? I would rather have the President focus on issues that really matter instead of thinking about driving, even if it's just down the street.
There are too many crazy people in this world that would try to hurt the President if he were seen out driving around town. This rule is certainly for his own safety. Imagine what could happen. People seeing him, ramming into him, shooting at him, etc. At least when being driven by the Secret Service, no one is completely which car he is in, let alone in which seat.
Cars are dangerous machines, something that the average person forgets despite accidents being quite common. The driver is usually the first to arrive to an accident and the last to leave. Also, the president makes a nice target. This does not change when there's a car involved; having the president at the wheel is a liability that would lead "accidents" surrounded in quotation marks.
No, the Secret Service is not being overprotective by not letting the president drive. The president can be driven around. He has too much on his mind to actually drive himself, however, he can be taken almost anywhere he wants to go without having to wait around for someone to take him.