Sensitivity represents how efficiently the earphone/headphone unit can translate the electrical energy into the acoustic sound. In general, In-Ear canal type earphones deliver higher sensitivity. That means an amplifier can drive them smoothly and efficiently without working hard. Higher sensitivity helps the Amplifier keep its best linearity and performance while saving the current consumption.
One the other hand, high sensitivity units may translate the system background noise into audible noise as well. Thus, for a high sensitivity unit, the system noise figure is more critical than the amplifier maximum driving capability.
Meanwhile, some headphones present very lower sensitivity and high impedance. To get enough loudness out of them, we need more Voltage from the Amplifier. Make sure we need the higher Voltage, not the higher Power.
For example, to SENNHEISER HD600, with 300-ohm impedance and 97dBSPL/mW sensitivity, an amplifier can deliver the maximum sound pressure level as below:
In general, high-sensitivity IEMs are built with very low impedance units, and high-impedance headphones provide very low sensitivity.
Below figures explain how much voltage and current are required for each target SPL(Sound Pressure Level).
To get the same 114dBdB SPL out of HD600, we need 40 x voltage level than SE846. Meanwhile, the difference in current flow isn't too much.
*Most smartphones offer the maximum output voltage less than 1V. Consequently, they hardly can drive those high impedance headphones up to enough SPL.
Earphones/headphones have a wide range of impedance. Thus, when we choose a headphone amplifier, the maximum Voltage of HPAMP would be the most critical specification representing its driving capability. In most cases, the Voltage can explain HPAMP's driving capability more easily than Power. We can't increase the Power to the load with a given impedance without increasing the voltage level. In other words, we can't increase the current flow without increasing the voltage level.
It's like a water gun. If given with a tube (impedance), the only way to increase the amount of water flow (current) is to push the piston harder(voltage).
If the current from the HPAMP isn't sufficient enough for the target voltage level, we will get the output saturated to the current limit. However, for the high impedance unit, the current flow isn't that much, and we usually have enough headroom in the amount of current flow.
In short, HPAMPs usually have enough headroom in current, and voltage matters in most cases. Meanwhile, Speaker Power Amplifiers, designed to drive 4~8 ohms load, require much amount of current flow and need to check both Voltage and current, i.e., Power in watt.