I had both - and they are virtually identical to my ears. If you want to add euphonic warmth to the mid-range, either EQ or go tubes. Love my 880s with the LD MKIV. Ironic thing is that I could have kept the 250ohm - but I had to try the 600's to see if there was a difference.
Might be slightly smoother top end on the 600s - but I realise that is probably placebo rather than fact .
I had both, but the 600 ohm was used and the pads were a bit flat. Don't know if that affected anything but I actually preferred the 250 ohm, though there wasn't a heap of difference. Even Tyll Hertsen said he thought the difference could very well be placebo--or wishful thinking perhaps, in many cases. There's also expectation bias and group pressure. When a number of people say the 600 ohm sounds better, hey, you want to be with the "in" crowd, right? And this is to ignore the factor of amp matching. Maybe the 600 ohm matches certain amps better.