Without cutting sections and analysing them under a petrological microscope, I have to make educated guesses, helped by some research on the Mount Mansfield area and Vermont geology in general..
1. As soon as I read green and greasy with black bits, I was thinking serpentine. That may be the case, but the shade of green is more like epidote and the texture less like serpentine. Both those, and also chlorite and olivine, are present in the area. We can't tell now because the specimen is detached from its outcrop, but I suspect it may have come from near the edge of a vein - where hydrothermal processes would have been at play. My best guess, partly because there are some notable mines of it in Vermont, is steatite (soapstone), which can come in a green colour. (Green Mountain Soapstone is the largest soapstone supplier to North America.)
Note: serpentinite is a rock composed of serpentine minerals; the two words mean different things.
2. Epidote can be nearly black, but in your specimen the dark mineral looks more like tourmaline. Again, I think this is probably from a vein system, hence the relatively large and distinct crystals. The white mineral could be quartz or apatite. An alternative is that it is from one of the instrusive volcanics in the area, in which case it could be gabbro, in which case the dark mineral could be pyroxene or amphibole (but I don't think there's much gabbro in the area, and what there is is more altered than in your specimen).
As I said: from photos of hand specimens, I can't be at all precise.