It seems to be time for a bit of clarification here...
The basic theory is that damping factor is the ratio of the output impedance of the amplifier compared to the load impedance posed by the speaker - expressed as a ratio.
This number matters for the following reasons.....
1)
If the source impedance of the amplifier was very high in relationship to the impedance of the speaker, then a lot of power would be wasted in the amplifier, and very little would make it to the speaker.
(This is a bit of a red herring since it hasn't been an issue with audio amplifiers since maybe 1920.... although it can still be a big deal with radio transmitters.... )
2)
When you look at it from the point of view of the amplifier driving the speaker, the source impedance ends up in series with the load (which is the speaker).
Because speakers are usually some sort of complex impedance, and the source impedance makes a voltage divider with the impedance of the speaker, this will cause some sort of interaction.
The exact result of this interaction will depend on the impedance of the speaker - but it will end up causing an error in frequency response.
The lower the damping factor of the amp, and the more the impedance of the speaker varies from a simple resistor, the greater the alterations (errors) in frequency response caused by the interaction.
So, if your amplifier has a very high damping factor, it won't interact much with different speakers......
And, if your amplifier has a low damping factor, then it will interact a lot, meaning that it will sound different with different speakers.
The interactions that do occur will produce much greater aberrations in frequency response with speakers that have more complex or "interesting" impedances.(In engineering terms, an amplifier is supposed to boost the signal without interacting, and an "ideal" amplifier has an infinite damping factor, so a low damping factor counts as a weakness in the design.)
3)
An entirely different issue (practically) is something called "back EMF".
In general terms, a speaker is an electric motor; current in the voice coil creates a magnetic field, which pushes against the field created by the magnet, and so the cone moves.
(Most speakers, excepting piezos and electrostats, work on this principle; in folded ribbons and planars the force pushes a flat diaphragm back and forth instead of a cone... but the idea is exactly the same.)
And, like every electric motor, a speaker is also a generator...... when you push on the cone, and so move the voice coil in the magnetic field, and voltage is generated.
If nothing is connected to the voice coil, no current can flow as a result, so this doesn't do much.
However, if the voice coil is shorted, current flows, and that current flow produces a force that opposes the movement that caused it.
(If you short circuit the output of a generator it acts like putting on the brakes.)
In a speaker, this "braking force" serves to control the movement of the cone - at least to some degree.
This obviously has a lot more effect on heavy drivers that have a lot of momentum (like big heavy woofers).
It has very little effect on folded ribbons and planars because, compared to a woofer, they have "very weak motors" (they have strong magnets; but not many windings in the magnetic field).
Basically, when you play music, the amp pushes current through the voice coil, which creates a magnetic field, and the "motor" pushes the cone.
Then, when the music stops, the output of the amplifier is at "0 volts", which acts as a short circuit on the voice coil, and prevents the cone from continuing to move (due to momentum).
The higher the damping factor, the more closely the amplifier resembles a true short circuit, and the more effective this braking action.
Another important issue is the design of the speaker itself.
Any moving mass, like a woofer cone, has momentum.....
Ignoring the amplifier entirely, the woofer cone has mass, and there is a spring holding it in the center of its movement range.... this is all mechanical.
And all cabinet designs include some sort of
MECHANICAL damping that tends to make the driver want to stop moving rather than bounce back and forth forever.
(This can include losses in the rubber surround, friction from air moving through fiberglas, and a bunch of other stuff.)
In a real world speaker, when the sound stops, there are several forces acting to get the cone to stop moving..... and they include
BOTH electrical and mechanical damping.
(All speaker designs are a compromise here; a lot of damping will make the speaker stop quickly, but also make it difficult to move to begin with, and so lower its efficiency).
And all this matters.... why?
Back when tube amplifiers were current technology, and all had low damping factors, speakers were designed with lots of built-in mechanical damping.
Now that most people use solid state amps, most with a very high damping factor, speakers tend to provide less mechanical damping, and to rely on strong electrical damping instead.
Older speakers, designed to be used with amps with a low damping factor, may end up sounding bass-shy if they're "too well controlled" by a modern amplifier - because they're overdamped.
And modern speakers, designed to be tightly controlled by the amplifier, and so designed with less mechanical damping, often sound sloppy when powered by an amplifier with a low damping factor.
(Basically because they're not being controlled very tightly by anything, and so tend to do their own thing, controlled mostly by their mechanical parameters.)
Note that, in most situations, this does
NOT cause significant distortion per-se.
(There is no specific reason to expect "more distortion when two tones overlap in time" as someone suggested... although it's possible that certain speakers may exhibit more IM distortion when not properly damped.)
When there is insufficient damping you will most often hear errors in frequency response, which will be very different with different speakers.
You will also hear the sort of time smear that results when the cone keeps moving after the music has stopped...
(You may hear this as "slow, sloppy bass", or, because the bass notes continue on for longer than they should, your brain may interpret it as "more bass".)
These are all generalizations, and the electrical and mechanical theory behind them can be somewhat complex when you look at it in detail......
For example, when calculating damping factor, the resistance of your speaker wire counts as part of the source impedance.
And, from the point of view of the driver itself, the impedance of the crossover counts as part of the source impedance.
And, from the point of view of how well back EMF is turned into mechanical damping, even the impedance of the voice coil itself becomes part of the source impedance.
(The resistance of the voice coil is part of the resistance the back EMF voltage has to overcome in order to cause current to flow.)
I would also like to discourage you from applying terms like "microdynamics" to this situation.
Terms like that seem to suggest that "certain equipment simply loses or forgets about fine details" - which is not strictly true at all.
That would be like saying that a certain camera lens omits certain small objects from photos taken with it.
In fact, it may blur them, but it isn't going to leave out certain specific objects.
You're talking about blur.... which is really a very different thing than "dynamics" (it's more properly considered to be "energy storage").
The other thing is that I've never heard an amplifier that "simply fails to amplify certain things".
It is the speaker that is causing a sort of time blur - which, in turn, may make it difficult for we humans to make out those details.
The amplifier's sole part in that process is in whether it doesn't control the speaker well enough - and so allows the speaker to "get sloppy" with the details.
(The amount of actual "time blur" in modern solid state amplifiers is absurdly small; thousands of times smaller than the mechanical time errors present in even the best speakers.)
I would also point out that electronics in general, and amplifiers in particular, are
VERY fast compared to speakers.
When audiophiles talk about "an amplifier being audibly slow" they're just being silly..... no amplifier introduces enough time error to the signal that human ears can detect it.
The example Gary provided, with a 50 Hz tone that actually fails to start and stop as quickly as it should, demonstrates an interaction between the amplifier and speaker.
(The amplifier itself responds electrically thousands or even millions of times faster than any mechanical device like a speaker.)
MECHANICALLY, the speaker acts like a resonator, and the amp is unable to "push energy into it fast enough" or "slam on the brakes fast enough afterwards" to accurately control its movement.
And, yes, in such situations, a more powerful amplifier, or one with a lower output impedance, may have more success.
(But I'm also inclined to say that the speaker design is problematic for requiring that much control to perform properly); this would absolutely qualify as "a difficult to drive speaker".
