In the same way I'd compare apples and oranges, although that surely has to do with the way I'm using the Anvil. I load it with exactly the amount I can pull in a single, hard pull--for me, that's a slightly under-packed half-oven. I treat it like an old-school bong rip. I've seen others load the full oven and take several pulls (or watched Whiff inhale the whole thing), or even toke on it native like a doob, but I think it shines best as a massive one-hit water piece. I often hit it while my DCE is warming up. I'll be really psyched to take it camping, where the DCE is just too fragile.
The DCE, on the other hand, is my clear reach-test winner. With a hard pull it can white-wall a water piece nearly as quick as the Anvil, or it can support a long, slow session with endless gentle draws. It pretty much does whatever I ask.
They both provide complete extraction (though the DCE likes a little stirring) easily. I believe the DCE is capable of more full-spectrum extraction, especially with some temp stepping, but very little is lost to the Anvil's heating process.
I love them both.