Yes they do IMO, and no they aren't (similar sounding).
Just my impressions ......
DT880 is a little dryer sounding, extends further (bass and treble), and handles faster music a little quicker. The bass especially is quicker (less mid-bass anyway), and they are a little brighter sounding.
HD600 is a little lusher sounding, They handle most music very well, but can sound a little congested with really fast music. They have more mid-bass, which can sound a little 'thumpy'. Although not as bright, they still have good detail.
The two things the HD600 have (again just IMO) over the DT880 is better timbre and a more believable sound stage & imaging.
I have both - and still use both regularly (along with my 325i and K701)
Interesting. And how do your 325i and k701 play into the lot?