Late to the party but just had the chance to get a HD800S, bought second-hand from this forum. I listen to folk/indie pop, hip-hop, acoustic, electronic, and some jazz and orchestral. I play the occasional first-person shooter as well. I'm running the headphones through my Centrance Hifi-m8 dac/amp, and my source is usually high quality MP3s via Spotify on my PC.
As there are so many reviews out there already, I'll just cover some of the biggest differences between the two headphones I personally experienced. I'm pretty new to the hifi community and don't come from a musical background, so I apologize in advance if I'm using some of my terminology incorrectly. However, I do come from a photography background, so I'll be making analogies there.
COMFORT
My ears were kept cooler on the HD800S as the earpads and dust cover never comes in contact and there's significantly less clamp pressure. The alcantara/suede like material feels nice against the skin, though some would prefer the softer feeling of the HD6xx's thicker velour pads. Only complaint here is that after extended listening sessions, there may be a bit of a hot spot on the top of my head from the headband where I feel the weight isn't distributed very well. In spite of that, the HD800S is more comfortable overall than the HD6xx and I found myself being able to listen longer without fatigue.
SOUND STAGE & IMAGING
Gaming with the HD800S, I can easily hear and pinpoint the direction and distance to enemy footsteps and gunfire while I found previously with the HD6xx, I often only had a general sense of direction to where the sounds were coming from, but the imaging was quite muddied in comparison. In music, the first thing I noticed is that the HD800S vocals are noticeably less intimate. The HD6xx often gave the impression that the signer was directly in front of me, almost whispering into my ear. In contrast, the HD800s feels as if I'm listening from two or three rows back. The benefit of this is that because the vocals don't seem to be so in-your-face, the background instrumentals come out a little bit more, but for certain vocal-centric songs (eg. "Where did I go" by Jorja Smith or "The Way I Am" by Ingrid Michaelson) I prefer the HD6xx's presentation a bit more. With instrumentals, electronic, and vocal jazz ( eg. Michael Buble or Frank Sinatra) I prefer the HD800S.
It's almost like the difference between photographing something from a bird's eye view (HD800S) vs a worm's eye view (HD6xx). With the bird's eye view, everything in the image is perfectly spaced and sized in relation to each other, while the worm's eye view will distort and emphasize certain elements (vocals).
FREQUENCY RESPONSE AND DETAIL
I should note that I did not hear anything "new" with my music through the HD800S, instead it brought out certain elements more. Most noticeably, there's more bass detail and texture and the treble holds better detail and "airiness". Everything also appeared to be separated better and less congested in the busier passages than the HD6xx as well. Starting thirty seconds into "ikigai" by Idealism, the bass notes while not more impactful than the HD6xx, sounds better defined and less one-note sounding. It's a very subtle difference, but it is there. In "Summer Montage" by Justin Hurwitz from La La Land, I could hear more of an sharper edge to the brass instruments, and the cymbal is more emphasized as well. If I had to sum it up, the HD800S feels more lifelike. You hear and feel the zing and harshness that comes from certain instruments, while keeping it extremely defined and separated from everything else that's going on.
With the HD800S, it feels like you're hearing the music very much the same way you would look at a raw, non processed high-resolution Gigapan image. When listening to the music is a whole, nothing is particularly emphasized, but as you "zoom in", all the details are still there and clearly defined for you to pick out and analyze, as you would in a 200 gigapixel Gigapan image. To hyperbolize, HD6xx in comparison feels like a single 24 megapixel jpeg photo run through a subtle VSCO filter that crushes the shadows and highlights but romanticizes what remains into something very enjoyable. While I don't have any experience with equalizers, I'd imagine the HD800S will probably respond better to it than the HD6xx, just as a RAW image file can be stand up to more processing compared to a jpeg file.
CONCLUSION
Photography is not about the latest digital cameras or its ability to resolve 50 megapixels. Sometimes, you get more enjoyment out of an old 35mm film Leica. In very much the same way for me, music isn't about the gear but rather the enjoyment of the soul and passion that comes from the music itself. The headphones are just a tool for that, and everyone's subjective opinion means preferences vary greatly. In my own experience, the biggest difference for me is the comfort of the HD800S, followed by its technical superiority in its sound. The HD6xx is still immensely enjoyable and is still my preference for some types of music, specifically for its intimate vocals. Both are extremely accomplished cans and I'm very lucky to own both.