Main differences between PIGMENTED black and DYE black?

I recently refilled a HP #58 photo cartridge, and the kit I purchased, from
Sams club, had CMY PM PC and K colors, the only 2 that were specifically
designated PHOTO colors were the Photo Cyan and Photo Magenta. The black (K)
was just- BLACK. So from what I understand, since its compatible with lots
of other types of printer carts, I'm assuming its a DYE based black, which
is pretty much what "photo black" is?

As I understand it, the HP #56 requires PIGMENTED black? However, when I
refilled my #56 I used this same ink. The results don't seem to be any
different than what I saw from the original ink in the cart- But I'm not
100% sure honestly.

So if the #56 (black) cart is supposed to be filled with Pigmented black
(which I can not find locally, I'll have to order online), and the photo #58
cart is supposed to have Photo black (dye based ink, which I guess is what I
have that came in the kits), what is the main difference? I know the
Pigmented blacks are much higher from the prices I've seen from online
vendors, but I don't know what the difference is.

Thanks again
-Tom