You want to know the difference between brown eggs and white eggs? NOTHING! Although brown eggs are often thought to be more healthy as they look more "natural", there is nothing more nutritious about brown eggs when compared to the white variety.
In fact, all eggs start out as white in their development and any color change after is simply due to genetic factors. In the end, as long as the chicken is raised under the same conditions, there will be no significant difference in nutrition or taste between brown and white eggs!
That said, brown eggs usually cost more because they typically come from larger hens who require more feed, and that cost gets passed on to you as the consumer. Want to save more money? Go with the white variety as apart from color, brown and white eggs are the same.