Since there is no legal definition, "premium diesel" means anything the retailer says it does. Some places, all it means is that this is diesel exactly like the "regular" diesel, but we charge 6 cents more per gallon for it to make you feel better. It might also have additives to increase the cetane rating, or maybe injector cleaners, or something.
My personal take on it is that if you ask inside and they drone running the register has no clue (very common, that), it is most likely the same stuff at a higher price.