the Carnaby's is the white-tailed black cockatoo of the wheat-belt and is the one that is commonly seen around Perth for part of the year. The Baudin's is the white-tailed black cockatoo of the karri forests and is not generally seen in Perth as it is much rarer and much more restricted in its diet. They are difficult to tell apart from one another if you're not familiar with them. I've seen both species at the Perth Zoo, but only the Carnaby's in the wild.
If I recall correctly they can be told apart by their calls, Carnaby's is "karak", Baudin's is "William", which I found hard to believe until I heard them in Dryandra, and that is exactly what it sounds like. Numbers are declining there because of introduced bees taking the best nesting sites.