Well, a cockatiel is a cockatoo. Then there's the traditional ones like the Galah and the Sulpher Crested which are much larger. But they could just be talking about the size difference in the different members of the cockatoo family.
I've only had limited experience with the Australian Cockatoo's - never had anything to do with one from overseas as yet. This site here www.australiancockatoos.com/ is a work in progress but has some info one some of the Aussie Too's, as I said it's a work in progress and I haven't had time to update the site for some time.