Impedance does three things for a headphone that I will mention.
1.) It is responsible for how much power the headphones draws from the source
2.) It damps the oscillation of the driver ( see electrical damping )
3.) It controls the phase of the signal
The higher the impedance the easier it is for a system to provide current to the headphones.
If the output impedance of the source is relatively high compared to the driven headphone the damping factor becomes quite low. This in turn may lead to undesired oscillations of the driver and thus noise / distortion.
So in many cases a headphones with 600 Ohms may sound very similar to its 60 Ohm version especially if a sources output impedance is low.
There are two numbers to remember with headphones.
1.) Sensitivity: How easy it is to drive the headphones. The higher the sensitivity the easier it is to make it the driver move.
2.) Impedance: How much current ( power ) the drivers and voice coils are going to draw from a source. If the source cannot keep up then a lower impedance headphone is better. If a source can handle it and damping is considered a higher impedance source is better because the damping factor goes up.