Ah one of the lifelong questions. Okay, basically white eggs are produced by white feathered hens, which have earlobes, and brown eggs are produced by red feathered hens, which have red earlobes. Brown eggs are more expensive, though because the chickens that produce them are fed more. Hope it helped.
There's really not much difference due to color, the difference really occurs on how the chicken is raised. A lot of eggs you find at farmer's markets are brown. These eggs are not mass produced and the chickens are often free range, able to eat grass and other natural goodies. These chickens will tend to lay eggs that, to me, taste better. The yolks are extra yellow and generally, just taste better. I know some Amish ladies that, in addition to grain, feed the chickens watermelon and other kitchen scraps, plus allow them to free range. Therefore, to me, the difference is not about egg color, but how the chickens are raised. I also saw at a local feedstore the cutest chickens that had feathers that looked like the markings of a chipmunk and they lay green eggs!!!!