The Hachiya variety of persimmon, when ripe, has a rich flavor and is very sweet and somewhat spicy. When unripe, the Hachiya is bitter and you will pucker like never before (it takes a long time to get the tannins - chemicals that are found in red wine, tea and other fruits - out of your mouth). So, always eat the Hachiya persimmon when very soft, and very ripe - the tannins become inert and the astringent values disappear when the fruit is ripe/very soft.
The Fuyu variety of persimmon can be eaten while firm, even green, and is a non-astringent variety. The Fuyu is smaller, somewhat flat, but has the brilliant orange color of the Hachiya.