The relative phase between the different frequencies. You're correct that the ear can't pick up a global phase change.

As an extreme example, consider a delta function: there is silence, then a brief spike, and then silence again. If you're just looking at the amplitudes of the various frequency components of this signal it is indistinguishable from white noise. The only thing that makes this signal look (and sound) different from white noise is the relative phase between the different frequency components. The ear's ability to detect these phase synchronicities helps it to pick out "peakiness" in waveforms more easily. (This is, in turn, important for understanding consonants in speech, which is extremely important for intelligibility, particularly in noisy environments.)