One possibility is that many amps have too much gain, so when you use them with lower impedance cans, the volume pot is always turned down low. Half the signal gets dumped back into the ground, etc. etc. But with 600Ω you get to crank the pot up higher, and less of the signal gets wasted. Also, some amps also only sound good when used past a minimum gain level.
Disclaimer: I own a DT880/250Ω, and have never heard the 600Ω version. Would be nice if the only difference is the volume, then all I'd need to do is lower the gain on my amp to hear what all the fuss is about.