The road bar is designed to be gripped all over the place, and experienced riders take advantage of that. Generally, the drops are used when aerodynamics becomes important on flats, descents, and short "sprinter" climbs. The lower position also improves bike handling on descents and corners. Hands over the hoods is a general-purpose position, because it allows quick access to the control levers, it's a little more relaxed and the diaphragm is more open, and the hands are in the right position for getting out of the saddle. Riding the tops opens the diaphragm more and shifts the weight back, so it's often used for long climbs. And some bars have a very useful spot on the bends just behind the brake levers.
If your road bike is giving you a sore back, look into getting fitted and work on the strength and flexibility of back, abdominal, and hip muscles.