If the two headphones have the same frequency response (magnitude and phase), and have zero nonlinearities at all (0% distortion, etc. and with some conditions met, they are pretty much textbook LTI), then from either the impulse response or frequency response, you have a perfect characterization of the behavior with any input. If those things are identical then they will sound perfectly identical with any kind of input
In practice because of various nonlinearities: think mechanical and acoustical resonances as mentioned above, maybe the nature of the damping and motion of the driver (I'm not sure really), and so on, those conditions aren't met. Thus the FR doesn't tell you everything.
Pretty much every potential behavior in the headphone, we could measure. There just may be an infinite number of measurements needed to describe the headphones, unless I'm missing some simplification.
Also keep in mind that different people have different ears and the sound that actually gets to the eardrums may be somewhat different from person to person. Additionally, there is some dependence on the positioning of the headphone on the head.
Edited by mikeaj - 2/28/12 at 4:26pm