The simple answer is no... Brown and white eggs taste the same, have the same nutritional value and are of the same quality. The difference lies in genetics. Hens that lay white eggs tends to be lighter in color and have white ear lobes. Hens that lay brown eggs tend to be darker in feather color and have red ear lobes. So, egg color really only has to do with the ear lobe color of the hen based on her genetics.
One difference people notice across all eggs, typically when comparing organic and conventional eggs, is that there can be a variation in yolk color. Organic eggs tend to have a darker color than conventional egg yolks. This is due to the diet given to the chickens. While the eggs will all taste similar, the type of diet fed to chickens can affect the yolk color.