This, phrased one way or another, is essentially what I found on every website about sugar, except for the C&H site, which, of course, said there is a difference:
"Because the sweetener is processed down to pure sucrose, there is no essential difference in taste or usage between cane sugar and beet sugar, and they are generally used interchangeably, except by the rare person who might be allergic to one plant source, but not the other."
As a matter of fact, I read that many vegans prefer beet sugar because cane sugar is filtered through cow bone char to whiten it. Beet sugar doesn't need to be whitened, because it's already white. I've never held the assumption that sugar had to only come from a sugar cane plant to be sugar, because I've always known that sugar also came from beets. Because it's called sugar cane doesn't mean that is the only place sugar comes from. The beets in question are also called sugar beets. There are also sugar snap peas, and as far as I know, we don't get sugar from them.