I always hear it in relation to something like "here are some screenshots for (upcoming game) to wet your appetite."

I always thought of it more as if you've had a little bit of something and your mouth starts watering and you're like "Damn, I am hungry now!"

Like appetizers.

Granted, I'm not arguing against whet being correct, I'm just saying the way it's often used is able to be taken in another way which would justify the other spelling :)