1. It keeps my helmet liner clean. A clean liner means I don't have to wash it, which means I never have to fight with getting it back in correctly.
2. It keeps my hair less tangled. Even in a braid, my hair has a mind of its own, and the bandanna makes a difference at the end of a long day of riding.
3. It keeps my neck warmer in place in cold weather. Pulling the fleece up over my mouth and nose doesn't do any good if it comes right back down when I put the helmet on.
A bandanna also serves as a backup dish sponge, a rag for messy emergency bike work, or a tourniquet if the day really goes to shit. I keep several with me on long trips.