I think I gave you the Straight Dope, guys. Airplanes are usually fueled right after landing. An airplane can build up a hellacious charge while flying through different air masses. You ground the plane to dissipate the built-up charge.
I don’t buy the fuel vapor from the overflow argument, because typically you ground the airplane at a point closer to the fuel vents than the actual nozzle hole is. But overflow vapor is certainly a really good reason not to smoke around airplanes. On a hot day, you can literally see the fuel vapor rising from a fully-fueled airplane as the gas expands.
The big worry is that you’ll open the tanks, and gasoline vapor will rise out of fuel filler neck. Then you touch the metal nozzle to it, a spark jumps across, and ignites the vapor.
ElvisL1ves is wrong to suggest grounding the nozzle, because the last thing you want is to complete an electrical connection right at the fuel nozzle. That’s why the grounding strap is far away from the filler neck - if a spark jumps, you don’t want it jumping anywhere near the fuel nozzle.
For the record, I used to be a ‘duty pilot’ on weekends, running our flying club. I’ve fueled hundreds of airplanes. I’m also a ground school instructor. Static potential buildup while flying is always given as the reason why the grounding strap is so important.
Cars don’t build up a huge static charge because the tires are generally good enough conductors to dissipate the charge, especially if they are dirty, oily, or wet (and they almost always are, to some degree). Airplane tires will eventually dissipate the charge as well, but if you fuel immediately after landing, the charge may still exist. But while flying, air friction against the body of the aircraft acts just like rubbing a balloon in your hair - ions are released and trapped by the airplane. And because an airplane in flight is not touching anything, the built-up charge has no place to go. Eventually, if the charge builds up to a great enough potential to break down the dielectric constant of air, it will dissipate back into the atmosphere in the form of sparks or ball lightning.
If you pay close attention to some airplanes, you’ll sometimes see little metal braided straps coming off a part of the trailing edge - these are static dissipators, to help the airplane shed its charge gracefully. Otherwise, you’d be hearing pops and clicks on your radio all the time. Some airplanes need them, some don’t.
Once you land, the remaining charge will slowly dissipate through the tires, but it may not be gone by the time you start fueling.
The risk of static discharge is even higher on non-metal aircraft. The newer composite planes are basically a big freaking capacitor, and can build up a large charge.