What is the difference between white and green asparagus?
Yes, there is a difference in taste. I think it's the chlorophyll, but I may be wrong. But the green asparagus has a "vegetable" or "grassy" taste which isn't present in the white asparagus. The white one has its own distinct aroma, which is less pronounced in the green one. 

As for texture, Caleb already said it. White asparagus is more tender, if you buy it fresh enough. On the other hand, non-tender green asparagus is snappy and somewhat brittle, like a normal stalk. Non-tender white asparagus is tough and stringy. When I cook white asparagus, I remove the lowest part (the most stringy one) and peel it (the outer layer is especially string prone) and sometimes, if it turns out to be low quality/old, it still has an unpleasant texture. The green one seldom requires removal of the lower part, and never peeling. 

Violet asparagus is like white one in both taste and texture, but not as common. 

Oh, and what Caleb says about both being the same plant is correct, but I have never heard of little shades. All farmers in Germany grow white asparagus underground. They make mounds of earth above the place where the asparagus is planted, and it has to grow a lot before it reaches the surface. They also cover the mounds with black foil to get the earth warm in spring. The harvest starts sometime in April and ends by tradition on June 24. It is done by hand, because machines would break the rods. This makes it a quite costly vegetable. And it should be eaten fresh, because the longer it spends outside of the earth, the stringier it gets. 

I don't share the common German enthusiasm for white asparagus ("royal vegetable" etc.) but think that it is a good vegetable in its own right. If you can get it fresh, it is worth eating now and then.