I had to turn the gas off temporarily, then turned it back on. Naturally I
forgot to re-light the water heater pilot and it was off for about 18hrs.
When I finally went to do that, I saw a puddle under the water heater. It
was dripping from what I believe is the drain valve. It's a white plastic
threaded fitting similar to the one in this pic and located in about the
same place:

http://media.rd.com/dynamic/88/77/84...er-2_size2.jpg

After lighting the pilot, the leak from the drain valve stopped, but then it
started leaking from inside. You can see and hear the tsssss tsssss as the
water droplets hit the burning gas element. I did not know enough at that
point to look for rust on the interior. Does this sound like a tank leak,
and if so, is it true that there is no way to fix it.

Why would the leak from the drain valve stop when I lit the pilot?