I have bought premium white (from Fabric.com) and it is a little different from the standard Kona white. I have some of each right here!
Premium does seem to have a tighter weave; the hand is a little different, too. It's thinner and more drape-y, if that makes sense. It feels like a bed sheet, actually.
About equal transparency between the two, but holding them both up to the light I can definitely see that premium has a higher thread count. It's a LOT like a bed sheet, actually. A good quality cotton bed sheet.
I've never bought the black premium so no idea if it's blacker than regular Kona black.