There is no difference in the nutritional value of white eggs versus brown eggs. What makes the eggs different colors is the color of the chickens. Typically chickens with white feathers and white earlobes will lay white eggs. Chickens with brown feathers and red earlobes will lay brown eggs.
Brown eggs are typically larger than white eggs so they are often more expensive but they are not healthier or better for you in any way. Other chickens lay blue eggs or speckled eggs depending on their color.