Photoshop's save for web function is rather limited, and gets is all wrong in terms of naming: the SFW PNG export only supports 8bit png files with 1bit transparency, while just about any other application supports 8bit with full transparency. Secondly, when you save a "PNG24" with transparency, it should really be called a PNG32, since it includes an 8bit alpha channel (8bit transparency).
Anyway, I am simplifying some of the concepts here. If you are interested in learning more, check out:
https://en.wikipedia.org/wiki/Portable_Network_Graphics
and
https://en.wikipedia.org/wiki/Color_depth
Now, the reason why PNG8 is not imported correctly, is simple: Toonz expects the transparency to be 8biy. When you save a PNG8 in {hotoshop, it results in a 1bit transparency which Toonz attempts to interpret, but fails.