The other way I've heard this called is "chapote prieto" by Spanish-speaking people
Edit: the fruits are unripe now, they will ripen to a black/purple color
" their taste is something like a prune, but less tart and more sugary" - http://www.wildedibletexas.com/2013/07/texas-persimmons-are-ripening.html
I have not had the pleasure of tasting one myself, yet, but if the squirrels leave some for me I will happily report back!
I've heard of people using these for jams and preserves, some people compare their taste (or maybe their texture) to sweet custard.
The fruits started forming in late winter/early spring and will finish up in late July to as late as October! The trees themselves (if they are female/make female flowers) produce fruit after reaching a maturity of about 6 or 7 years.