I think they're fairly similar. The DT860 might be slightly more "commercial" sounding, with more midbass and energetic midrange.
Personally, I prefer the DT880, but ONLY if you got a good amplifier/source for it. If you don't have an amp, or will only be using a soundcard/MP3-player, the DT860 is a better alternative. It's easily powered, and not so fussy about sources. Make no mistake, the DT860 will also reveal a bad inboard soundcard, but not as ruthlessly as the DT880.
The DT880 is more comfortable, but the DT860 is by no means uncomfortable. It 's just a little heavier, and haven't got the same ultrasoft velour pads.
The DT880 is much more solidly build. It's more metal and industrial plastics, where the DT860 is all-plastic, and got that dreaded headband. The DT860 isn't bad, it just feels cheaper. However, the DT880 would survive WWIII, the DT860 wouldn't.