The simple answer to this question is that, beyond the color, there is no difference between chicken eggs of different colors. White and brown eggs are nutritionally identical, and, on the inside, they are physically identical as well. Despite what you may have heard about how one is more natural or nutritious than the other, both colors of eggs are products of real, live chickens, and their exterior color is not an indicator of nutritional content or anything else, beyond the breed of chicken involved in egg production.
One may well ask why there are white and brown eggs at all, if they are nutritionally identical. The color of chicken eggs is determined by materials which are deposited while the eggs develop inside the hen's oviduct. Some chickens deposit white pigments, while others deposit brown pigments, and some chicken breeds like the Aracauna and Americauna lay blue to green eggs, just to add to the color spectrum. The original predecessor of the chicken, the Red Junglefowl, lays cream-colored eggs. Different egg colors appear to have developed over the course of centuries of breeding.
Both white and brown eggs contain a yolk and a white, each enclosed by a protective envelope, along with a tough outer membrane between the white and the shell, and two chalazae to keep the yolk anchored in the egg during fetal development. When you crack an egg open, you can get a number of clues which will provide information about how fresh the egg is, and how nutritious it is.
In fresh eggs, the string-like chalazae are clearly visible and well-formed. More nutritious eggs have darker yolks, reflecting a greater concentration of vitamins and minerals. Since the yolk is supposed to feed a chicken while it develops in the egg, yolks are already designed to be highly nutritious, but the more nutritious, the better, in the eyes of many consumers. Eggs with dark, firm yolks and distinct chalazae are fresh and highly nutritious, no matter what color their shells are. Fresh, nutritious eggs tend to taste better and perform better in baking.
While it's not a hard and fast rule, in general, chickens with red ear lobes lay brown eggs, while chickens with white ear lobes produce white eggs. Ultimately, white ear lobed chickens will lay white eggs most of the time, it really depends upon the genetic makeup of the chicken. White chicken eggs were pushed on consumers for a brief period of time in the 20th century, leading some people to believe that all chicken eggs were white. This practice was presumably designed to make it easy for consumers to see that the eggs were spotlessly clean, to allay concerns about food borne disease. When farms began selling brown eggs again, many promoted their products as “natural” in the hopes of appealing to a specific customer demographic, when in fact white eggs (and green ones) are just as natural. White and brown eggs are usually sold separately, reflecting the fact that most egg farmers use flocks of one breed only, but eggs from small farms often come in a mix of colors and sizes, reflecting a diverse flock.