Distortion and compression are quite different in speakers.
Compression occurs due to voice coil temperature altering the overall efficiency (and spectral output) of the drivers. A driver that was efficient when cold, will be less efficient when hot. The base level distortion inherent in the driver may vary little during that time. Voice coils, magnet structures and crossover components are all susceptible.
In smaller speakers, the effects of compression are enormous in my experience, up to and including the levels (and duration) required to introduce significant audible THD.
If the so-called "preference" ratings among speakers are done at low (80s) levels, they aren't worth a red-cent.
That's probably the condition most readily and commonly associated with "compression". But I've also seen the word "compression" used in any context where either the amplifier or the speaker has begun behaving in a non-linear manner, which is to say, the gain (or effective gain for a speaker) is less at signal peaks compared to low signal level. Notwithstanding that the word "compression" is used to describe loss of efficiency in a way that does not necessarily imply non-linear distortion, I suspect that this is a word use that evolved from a slightly earlier word use where the word was used expressly because it implied dynamic compression of the signal, i.e., non-linear behavior with the accompanying non-linear distortion. I'm just guessing about this, but it makes perfect sense to me, because the word obviously does connote signal compression.
Even if this isn't a correct guess as to the history of the practice of using the word "compression" to refer to temperature-related compression, it seems pertinent to allow the possibility that temperature might possibly rise and fall with sufficient speed that there would be signal compression and non-linear distortion even in cases where the primary, most notable effect is a loss of efficiency. I'm not at all certain about this, but it was brought on another thread here a couple or weeks ago, in a thread concerned specifically with the effect of high temperature on the behavior of a loudspeaker. The instantaneous rate at which electrical energy is converted to heat will vary as V^2, but because heat is also being carried away from the coil at some instantaneous rate, the instantaneous rate of temperature increase is not a perfectly linear function of the power.
An interesting aspect of this question is that as temperature increases and electrical conductivity decreases, thermal conductivity increases. The Wiedemann-Franz law is a sort of odd duck in that the ratio of the two conductivities, the ratio of thermal conductivity to electrical conductivity, is proportional to absolute temperature. The constant of proportionality (the Lorentz number) is the same for all materials, i.e., even though the two conductivities are different for different materials, the ratio of the two conductivities is the same for all materials. The practical implication for speakers and amplifiers is that as temperature increases and electrical conductivity falls in inverse proportion with the absolute temperature, thermal conductivity increases in direct proportion with the absolute temperature. This introduces a twist that would obviously need to be taken into account in any predictive model of how temperature alters the behavior of an audio component, given that heat transfer is not immediate (it has hysteresis) and has to be incorporated in the model, in order to accurately model the way in which temperature depends on V^2.
Ah, I almost forgot to mention one example (context) where "compression" alludes to power compression, where non-linear behavior is unquestionably implied, and where power compression isn't due to temperature rise. Somewhere in one of the Klippel papers where various kinds of loudspeaker distortion are analyzed, there is a brief analysis of the power compression that occurs in the port output of a ported speaker. The explanation for why power compression occurs has to do with the fact that the resistance of air flow at the inner surface of the port cylinder is proportional to the square of the air velocity. Resistance to air flow is consequently greater at high loudness and at signal peaks, thus implying signal compression. As such it is referred to as "power compression", which suggests that when you talk about compression you're supposed to make it about power for whatever reason. Whether you make it about power or pressure, it is signal compression that implies non-linear behavior and therefore non-linear distortion.