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.
Martha's Vinyard Magazine suggests that the diamater of the stalks is not directly related to their age as one might suspect:
Some might assume thinner spears are younger and therefore more
tender. The diameter, however, has more to do with the age of the
plant itself and the particular asparagus variety.
They go on to cite Cook's Illustrated, saying:
[T]hin and
thick spears as equally sweet, nutty, and grassy, but ...
thicker spears have a slightly better, more tender texture. The reason
is that the fiber is somewhat more concentrated in thinner spears.
You can peel asparagus if you choose to. Personally, I peel the base of thicker stalks. Mark Bittman recommends peeling from just under the flowers to the base of the stalk on the thicker stalks.
Thinner stalks may be easier to saute or grill, and thicker ones more suited to steaming, poaching, and so on, but this is more related to the time it takes them to cook through.
See also:
Best Answer
I've heard the same recommendation before - the idea is that if asparagus is very thick (say, 1/2 inch or more in diameter), and if the bottom is much thicker than the top of the spear, then peeling it will help it to cook more evenly.
If you have particularly thick asparagus that seem tough, you may want to peel the bottom section. First, chop or snap off the bottom inch or so, as it will most likely be inedibly tough. Then peel the next 2-3 inches up. You'll want to lay the asparagus down on a flat surface to do so, so as not to snap it while peeling.
I've never had asparagus from a grocery store that was tough enough to warrant this, but it can happen with homegrown, for whatever reason.