Here's what I got back from fabric.com:
"Of the two fabrics, the "Kona Premium Cotton Jet Black" is the darker material. "

That's hysterical...and so not worth the extra cost! I'd be asking Robert Kaufman exactly what it is that makes the price higher for the premium. "Darker black" just doesn't qualify as a reason when the regular Kona Solids come in over 200 colors already. I think you got brushed off. :hunf:

My understanding also is that it is double-dyed.

The same as over dyed fabric cost much more? My experience with over dyed or double dyed is that it just bleeds more so the extra price and color runs down the drain.

Then why is SouthWest Decoratives claiming it has a higher thread count? And is the Ultra White also double-dyed?

You will notice that not all black fabrics are the same. The reason is that many blacks are simply overdyes of other colors. Kona jet black is made to be black from the start, and is not an overdye of other leftover fabrics/colors.

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.