How does experience relate to volunteer assignments?

It's both. Positions are assigned as people sign up, generally in batches every couple weeks. So the person doing the assignments looks at a batch of maybe 10-20 people, says, "What do these people WANT to do, and what CAN they do?" and then assigns them.

Signing up early means that if you're a possible fit for some particular spot, you might get it. Signing up late means that the spot you want might already be full.