I haven't done extensive research on the differences between the two. I have heard people claim that the white ones have been bleached, but I haven't found evidence to back up that claim, so I don't believe it to be true.
The real reason for the difference in color is the mineral content. My cousin's chickens lay all sorts of colors. Some are blue-green. The majority of the shell is comprised of calcium carbonate, but the other trace minerals determine the color.
I have compared the nutrient levels of both the brown and white eggs for Eggland's Best and both contain the same amounts of all of them, so I buy the white ones because they are cheaper...