My comparison isn't exactly the most valid since the closest thing to the Apogee Groove that I've listened to is the Audioquest Dragonfly Red, but here's my lowdown:
As soon as portability goes into the equation, something is almost always sacrificed. The Groove is a great option for those who would like a portable option to travel with.
However, in pure sonic merit, for me the Dragonfly, in comparison to a CTH, sounds tinny, wimpy, and underpowered. Bass loses texture and can become kind of flabby. And yes, I use the HD6XX as one of my main headphones to audition gear (I'm familiar with its sound). So I can't imagine the Groove being that much different.
Keep in mind that impedance in dynamic driver headphones is NOT purely resistive; it "swings" up and down. In the case of the HD6XX, the swing can be up to 500 ohms. That's a pretty hefty load the amp is going to be under.