You are referring to the difference between astringent vs non astringent persimmon. Basically all American/Common Persimmon (Diospyros Virginiana) are astringent type (Morris Burton may be debatable), meaning though they may be sweet, you need to wait for the tannins to breakdown by waiting till they are soft (or accelerate tannin breakdown artificially through CO2 or some other mechanism). You can eat them still hard, and while they will taste sweet, they will also be very tannic/astringent/fuzzy mouthfeel.
Also, there are two different species being discussed in this thread. The Diospyros Texana is a blue/black skin persimmon with blue/black flesh. It’s about 1 inch in size, like a berry. The one pictured with orange flesh cut open is D. Virginiana. The blue/black skin is simply a morph. They don’t fundamentally taste or behave differently than the “regular” D. Virginiana.
I was just sent some blue skinned D. Virgiana fruit, picked from Don Compton’s place. @jcguarneri @luxin Most of them are heavily seeded but that’s probably because Don has so many persimmon trees. They are ok imo. As @SMC_zone6 noted, they still fall behind the D. Virginiana x Kaki hybrids like Nikita’s Gift in flavor, size, texture, etc…