Why can I resuse green tea leaves several times, but not black?
I've never had an issue reusing anything that's real tea (black, oolong, green). Perhaps there is another factor here? If you're black tea is coming out of a tea bag, or a lower "grade" then the green tea leaves you're using that might be a reason that it isn't coming out desirably.

Generally speaking it's an issue of grade rather than the type of tea. The finer the tea, the more likely you are to extract everything in the first run. Take apart a cheap tea bag, and you'll notice that the leaves are powder or almost powder. A full leaf (you can buy any black tea from a reputable shop in a higher grade) will extract more slowly and give you a better flavour and probably yield to better second or third runs.