Technically, I think what's missing there is "gain" rather than just high voltage. Most modern music likely won't require an insanely high voltage to get the DT880 600 to play loud, but the built-in amp of mobile devices typically have very low gain, so they won't be able to get the voltage at output to play loud enough.
For instance, let's say... phone has a built-in gain of 2x. One song plays at 1Vrms, so the DT880 will receive 2Vrms at output. Technically should be enough for about 70-80dB listen, I think (just rough estimations here, no actual calculation was done). Another song plays at 0.5Vrms, so the DT880 will receive only 1Vrms here. Think those Youtube videos with the audio attenuated to hell.
Those are the times that an amplifier will be necessary IMO. Otherwise, the DT880 600 should technically be less susceptible to noise and other stuffs that the 32 Ohm version would otherwise be sensitive to.