Almost all amplifiers are voltage-drive amplifiers
Designers and manufacturers put a lot of effort to create low distortion amplifiers. Almost all amplifiers are voltage-drive amplifiers. They translate the input voltage to output voltage, with the least distortion possible. Voltage-drive amplifiers control the output voltage. The output current depends on output voltage and impedance.
A loudspeaker does three things
If an electrodynamic loudspeaker is driven by voltage, the loudspeaker translates the voltage into current. It does this with some distortion, because the impedance of the speaker is not linear. This distortion is called current distortion.
An example. Crossover designers use air core coils, because coils with ferromagnetic materials would color and distort the sound. But one forgets the biggest coils of the loudspeaker, the voice coils! The voice coils interact with a ferromagnetic material, like ferrite or neodymium. This gives a non-linear impedance, which distorts.
Next, the loudspeaker translates the current into sound. And it does this also with some distortion. So a distorted signal is distorted again when translated to sound.
A loudspeaker acts also as a microphone. It translates the motion of its diaphragm into a voltage. This translation introduces distortion. The microphone picks up all sounds, not only the, already distorted, self created sound, but also noise inside the enclosure, sounds from other speakers and noises and vibrations of the speaker itself. The generated voltage disturbs the voltage and creates additional distortion in the current and in the sound. You may think that the microphone effect cannot have a great effect, but at the speaker resonance the microphone effect reduces the output with 90 till 95% (and is called electrical damping).
With voltage-drive the voltage is controlled and the current is distorted, and the current translates to sound.
Lower distortion with current-drive
With current-drive the amplifier translates the input voltage to an output current. A current-drive amplifier controls the output current. The loudspeaker translates the current into sound, with some distortion.
The speaker distorts the relationship between current and voltage. But the current-drive amplifier controls the current, so there is no current distortion involved. The amplifier translates the input voltage to an output current, so there is no implicit voltage to current conversion via a non-linear impedance.
The voltage of the microphone effect does influence the output voltage, but not the output current, because the output current is controlled by the amplifier.
With current-drive the current is controlled and the voltage is distorted, but (only) the current translates to sound. With current-drive all current distortion by the implicit voltage to current conversion and by the microphone effect are eliminated, which results in much lower distortion levels and a much clearer sound than voltage-drive.
Tube amplifiers are not current-drive amplifiers
Quite some people think that tube amplifiers operate in current-drive mode. But because tube amplifiers have an output impedance in the order of the loudspeaker impedance, they are more in power-drive mode, which only halves the distortions, which current-drive totally eliminates. Current-drive amplifiers have a much higher output impedance.