Vegans prefer to not have refined white sugar made because it was ground up with animal bones, kinda makes the cupcake look less appetizing. In North America most white sugar is made from beets, and use the bone char method. Brown sugar is just white sugar with molasses in it, still bone-y. Some cane sugar is unprocessed and raw and therefore didn't get ground up and whitened with bones, so they will eat that.
Lacto ovo vegetarians do not worry themselves with the contents of sugar, so all is safe.