Exactly. The Dyson sphere's and shells merely surround a star to absorb the radiation emitted by the star. An astrovore, or whatever name you want for a "star-eater", is doing something to actively "feed" off of the star itself. Maybe it's using a companion body to syphon off hydrogen & helium to fuel something, or it's dumping in quantum black holes to increase hawking radiation to suck up, or something so exotic we haven't even imagined it. It seems very likely that whatever they are doing, they also have a dyson sphere or shell to aid with the system, but as this is all wild speculation, we are just shooting in the dark... at stars. :D
If you want a description of the difference that's a little more down to earth, then imagine a media star, music, movies, whatever.
Now the dyson sphere or shell is like a bunch of fans surrounding the star, listening to every word, ogling their appearance, and just generally basking in the glory they imagine is there.
The Star Eater on the other hand, is right up there hands on groping the star, stealing whatever souvenirs they can pull off, and possibly even trying to draw blood or cut hair to keep. Whether or not they have a possie of cronies surrounding the star to keep things "convenient" for them doesn't make it the same as the previous example despite some similarities as the intent and method is very different.
I hope this example was funny and somewhat disturbing. If it wasn't, you crawled out the wrong tube today. :P