The DT880 is still a typical Beyerdynamic and has the annoying treble spike (the 600ohm less so). If you don't mind that, more power to you, I can't stand it. I agree the DT880 is tighter and the bass is more extended but the treble spike makes it inferior to the HD600.
Actually the 600 ohm has pretty much the same treble spike. And yes, it's annoying. Trouble is, apart from that the 880 is one of the great values and one of the most neutral cans on the market. On acoustic music it can match it with the big boys--once the treble spike is addressed.
Fortunately that spike is at quite a high frequency, making it vulnerable to attack. I use a simple tone control on an integrated amp. Others use more elaborate EQ. Some use warm tube amps, which more masks than addresses the problem. Never mind, whatever works. Of course you can just ignore the 880 altogether, but in that case you'll be ignoring a potential gem, especially if you're into classical or jazz.