I have only spent brief time with either headphone, but IMO, it comes down to this:
HD600 has a very "even" sound coming from most sources I tried it with (at the SF Bay Area Meet). But without the right source, the low-end is just barely there, the top end is not very sparkly, and the midrange is just so so (unexciting). That was out of my Fiio E12 (which I no longer own). I guess I don't even need to describe how it sounds like out of my iPhone.
Now... that HD600 wasn't mine, so I couldn't just take it and plug it into the Bottle Head Crack a couple of tables over. But what I could hear of the Bottle Head Crack was superb clarity, smoothness, space, and resolution, with some added hints of warmth and weight. It sounded marvelous with the HD800, and I don't doubt that it would be able to deliver the HD600 much the same way.
I now own a W1000X, and from my brief time with it ("brief" because I haven't gone past 10 hours of head-on time), the sound is lush, smooth, and bright, but slightly warm, and with good impact even unamped straight through my iPhone or MacBook. The W1000X unamped sounds significantly better (more dynamic, more spacious, clearer) than how I could remember the HD600 sounds like. And it's less fatiguing than most HD800 that I have heard thus far. That said, though, I don't think my Leckerton UHA-6S Mk.II does much to the sound (other than making it clearer and smoother), so I'm thinking of some tube amp now. The W1000X sounds like it's craving some serious tube amping.
I must say... my personal preference tends to fall toward the W1000X for the coloration in the midrange, and because its tonality is an exact match with my favorite signature: the ATH-ES10. What can I say, I'm a midrange-head! I used to think that I was a bass-head, and even a treble-head at times (when I used to own a DT880), but now I know I'm a midrange-head at heart, and thus... Audio Technica is my signature of choice. To me, this is "neutral".
That's to say, I went into this comparison already with a bias, so... please take it with a grain of salt. But what I can say definitely is that unamped, the W1000X does sound much better than the HD600.