Speaking as someone who grew up around farms and helping out on his uncle's farm, there is no difference except that foolish people who grew up in cities will pay more for brown eggs and then swear there's a difference so they don't have to admit they paid more for no reason. Also, the breeds which lay brown eggs tend to be larger birds that eat more and therefore cost more to raise, which increases the price of brown eggs slightly, but not enough to account for how much more brown eggs cost in supermarkets.
Now, locally-raised eggs, whether white or brown, will generally taste different than commercial eggs because locally-raised chickens tend to have a more varied and less bland diet, which affects the flavor of the eggs as well.
ZV