Vanilla is what you get if you go to a run-of-the-mill ice cream stand, or buy it in bulk. It's not the most smooth vanilla, not the grittiest, and it's flavor isn't overwhelming, since it just uses vanilla extract. It's extremely average.
Vanilla Bean is a more gritty vanilla, since it contains the bean, and the very tiny black specks you find in it are minuscule pieces of the vanilla bean. It's much more flavorful than plain vanilla.
Double Vanilla is a mix of Vanilla and Vanilla Bean, I think.
French Vanilla is made from a custard in addition to either vanilla extract or vanilla bean. It's a lot more creamy than the other vanillas, though its flavor isn't as prevalent as Vanilla Bean.