They don't. It does change the sound and in making it more lean and etchy it can seem to be more detailed. Its not.This isn't how it works.
What's going on is how the amp interfaces with the speaker, how the amp behaves when its driving a higher impedance.
And how the amp behaves is that it will make less distortion, and this will be heard as 'more relaxed more detail'; IOW neither 'lean' or 'etchy'; quite the opposite.
A tube amp will make less distortion (assuming that it has a 16 ohm tap if using an output transformer) and so will have less of the lower ordered harmonics. In this way it will simply sound more neutral. Because there is less distortion, there will be less to mask detail.
A solid state amp tends to not have much of the lower ordered harmonics- their distortion signature tends to be more of the higher ordered variety. But again, they will make less of it, so they will sound sound smoother (since these harmonics are interpreted by the ear has brightness and harshness) and more detailed; the latter simply because there will be less distortion to mask detail.
So we see that in both cases we have the same benefit. Now if its between 4 ohms and 8 ohms, the 8 ohm solution will have lower distortion than the 4 ohm. Again, less distortion (though not as low as with 16 ohms). This BTW is easily seen in the specs of any amplifier. Being that we really are talking about high end audio, where the goal should be to get the music to sound as real as possible, lowering the distortion should be part of that solution :)
As far as the speaker cables go, its a simple fact that at 4 ohms, the DC resistance of the cable can play a role in the damping available to the speaker. The same speaker cable on a 16 ohm speaker is simply going to have a negligible effect on damping.
Now this depends largely on the assumption that the higher impedance speaker is simply higher impedance, and otherwise has the same breakups and other anomalies that it would have if 4 ohms or 8.
If your speaker has level controls or level adjustments, they are there for one reason only- the voltage response of the amplifier is unknown, and the control is there to allow the speaker to be adjusted to that voltage response. This is indicative of the speaker being built for amps that behave as power sources rather than voltage sources (for example, any tube amp that runs zero feedback). For more on this topic see:
http://www.atma-sphere.com/en/resources-paradigms-in-amplifier-design.html