What's the skinny on fat asparagus?
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: 


Mark Bittman writing in the New York Times
Better Homes and Gardens
Martha's Vinyard Magazine