Yep, it's pretty common for those two names to be used for the same coral. I've also seen a couple places call them torch corals, which can get confusing since there's a completely different coral that's also called a torch coral.
ya i did some research too, and i guess trumpet corals end up poofing up more, making it look like one larger coral, whereas the candy cane looks more like individual polyps
This is exactly why I hate the use of common names, "Caulastrea" at least ensures I know what genus of coral is being discussed, if the actual species is unknown, then simply Caulastrea sp. at least keeps us all on the same page. But what really starts me banging my head on the desk is the use of fad names such as "shield of light" and "superman"....aaaaahhhhhhhhhhhh!!!!! Talk about dumbing down the hobby. Sorry, am ranting.
While I put this page together for ID purposes, I have also found that its a good study guide in getting me into the habit of using proper genus names also.
I think that the trumpets are the Caulastrea curvata and the candy canes are the Caulastrea furcata. But that doesn't mean that they aren't mismarked or the names tossed around as if they were the exact same thing.