What happens to Lilypads in the rain?

If the circuit is powered when you put it in water

Water would add random paths where the current can flow in your circuit. Most probably something will short out. If you're lucky, the circuit will just not work, in the worst case you can damage the chip on the Lilypad. If you predict that your circuit will have contact with water, you can experiment with:

• enclosing it in something waterproof (some synthetic material, maybe?)
• using waterproof fixatives from art supplies (these are in spray and coat your electronics with a thin waterproof film of chemicals). There are also some specialty products for coating electronics, although I haven't tried them.
• to make your LilyPad really safe, you can cast it into epoxy resin or something of the like.
• be sure to insulate every exterior wire, too. (For example wires that go to an LED somewhere)

If the circuit is not powered when you put it in water

It shouldn't be a big problem. The circuit will be okay AS LONG as you let it dry completely before powering up again. Most cell phones can survive trips to the bottom of the lake (the batteries don't). It is a good idea, if you consider spontaneous swimming, to put the battery out. The problem you may have is the circuit rusting prematurely, but that shouldn't be the case of arduino and lilypad.

Not owning one myself, I'd need to guess.

According to the arduino website, the pad itself won't get harmed:

Washability
Wash at your own risk - we do ;). We recommend washing projects by hand with a mild detergent. Drip dry. Make sure you remove your power supply first!

Electronics and water don't mix usually for two reasons:

• short circuits: the ions in the water short circuit the board, rendering it infunctional or even destroying it. The latter happens when power from the supply voltage goes to places it shouldn't go. This is especially important for high voltage devices.
  • rust: water reacts with certain materials and corrosion occurs over time

I'd guess that a coated PCB designed to be washed won't rust (that easily) and probably also don't react to short circuits that easily. So in rain there is a good propability that it will work as expected. Submerged under water, however, I'd expect it to fail (primarily due to the open pins on the outside) and you should specially protect your power supply and leads. (hence the warning in above quote)

You have to make similar reasonings about the other components of your clothing. If they are designed to withstand washing they'll probably work in some rain, too. But only really waterproof connections will work with that much water you have when swimming.