When you hit the supermarket to grab some eggs there are the brown ones and the white ones. They're different colors and the brown ones cost more, so obviously they're different, right?
While there is a price difference between the two colors, that's about all the further the difference tends to run, according to the folks at the Today I Found Out YouTube channel. The difference in egg color is simply genetic. Chickens with white earlobes tend to lay white eggs, while chickens with red earlobes, which generally have red or brown feathers, lay brown eggs.
Part of the difference in cost does come down to a perception of brown eggs as somehow being better, but it's also in part because those red and brown chickens are a larger breed of chicken. That means they require more feed than white chickens to produce the same number of eggs, increasing the cost per egg for the farmer.
Additionally, as C. Claiborne Ray wrote in the New York Times in 2014, there's no evidence of any nutritional value difference between the two colors of eggs, outside of brown eggs having a marginal increase in omega-3 fatty acids. In fact, research suggests there's no difference in the nutritional content of eggs regardless of color, grade or how they're raised (organic, free-range, or conventional). There's also no difference in the thickness of the shells. Shell thickness tends to be a product of a chicken's age, with younger chickens laying harder eggs no matter what color those eggs are.