The is no difference nutritionally.  It is believed the color difference comes from the color of the hens.  Hens with white feathers and white earlobes tend to give white eggs while hens with red feathers and matching earlobes give brown eggs. 

Hens that give bown eggs tend to be larger than the hens that give white eggs.  Therefore, they  require more feed and care, and the extra expense is passed on to the consumer.