Owning both, I'd disagree a little in terms of neutrality.
Neither is spot-on neutral, though both sound very natural. I'd say that the HD 600 is the more neutral of the two, with having a moderately warm and bassy tilt. The HD 600's upper bass is, though, always there as it's somehow got a peak and rolls off more than the HD 800 which has, on the other hand, overall a tad more bass than the 600, but extends lower, making the moderately emphasised bass less obvious. One thing you notice when listening to the 800, compared to something more neutral (ER-4S for example), is that bass also bleeds into the ground tone a little, but way less than Audeze cans.
The HD 800 even has got a moderately v-shaped sound signature which comes in handy when one is listening at low volume levels as our hearing will perceive a moderate v-shaped sound as more neutral at low levels (keywords: loudness, aurally compensated volume). Sennheiser even somehow recommends to listening at rather moderate volume with the HD 800 in the manual somewhere.
In terms of resolution, instrument separation and speed/attack, the HD 800 wins, hands down. Though, both have got a pretty different sound-stage presentation: the HD 600 stage is rather small, compared to the HD 800, but has got a balanced amount of width and depth (all in my ears and from my perception, of course), whereas the HD 800's stage is darn wide and, though it has some decent depth, imho lacks about 60% of depth to create a balanced ratio of depth too width.
Instrument separation is very precise and sharp with the HD 800 which is also able to generate (subjectively perceived) "black, empty space" between instruments.
The HD 800 is less forgiving than the HD 600.
Imho, though the HD 800 is definitely Sennheisers' flagship out of the current models, providing stellar flagship sound, the HD 600 is Sennheisers' model with the best value-for-money ratio. I think there's a reason why they're still producing them (though they seem to have slightly changed over the years), just as the HD 25.