This exact question was discussed in a human physiology class that I
took in the 1960s. Basically, the answer is that only the outer
surfaces of our fingers and toes absorb water by osmosis, and expand,
while the layers beneath are waterproof. This results in the outer
skin being temporarily more ample than its underpinnings. It's as if
you are wearing a pair of socks that are much too large for you:
wrinkles result from the lack of tautness. Once the temporary
waterlogging has ended, and the top layer of skin returns to its
normal contours, the wrinkles disappear.
"The top layer of the skin is composed of toughened, scaly cells
collectively known as the stratum corneum. On most of the body, this
layer is quite thin, just .015 of a millimeter, but it's 40 times as
thick, or 0.6 of a millimeter, on the soles and palms.
Normally the stratum corneum is relatively dehydrated, but it absorbs
moisture and swells up when soaking. This swelling occurs throughout
the soles and palms, but it's most noticeable in the fingers and toes
because of their restricted dimensions... Since the underlying tissue
doesn't absorb water, the stratum corneum can't spread out and it
buckles like asphalt on the highway in the summer sun."
The Straight Dope Archives
Here you'll find some more explanations of this phenomenon:
Scientific American: Ask the Expert
Department of Biology, Clemson University
MSN: You Asked for It
This is the combination of search terms that brought me the best
Google Web search: bath water wrinkle fingers
If anything's unclear, please request clarification. I'm always
willing to re-ponder an interesting question like this one.