quality over quantity for some cans,essential drive requirements for others.
Using portable systems as an illustration :
obviously the headphones/buds that come pre-packaged with portable players would not benefit one bit from being amped requiring neither the power increase or increased detail which not only would not be a help but can even be detrimental in that it imposes anye shortcomings on the listener.no escape
Then there is the full size headphone used for portable duty case.that is where a headphone from either being inneficient or due to impedance mismatches needs an interface that will allow the front end source to drive the mismatched cans and that normally mean high imdeance input/gain stage to increase voltage levels to a usable state/low impedance output headphone driver.
Fail to have the proper device between the source and headphone sand you can expect not just reduced volume levels but what volume level you do get will mostly be totally lacking in dynamics,will be bass shy,have zero low level information retreival and more often than not will clip the output section on peak music events.
Esential gear for the 200 ohms up headphones
Upgrade those "generic prepackaged" cans to mid level high efficiency headphones that can be driven easily by a portable and it gets a bit more cloudy.Far less of an "absolute" unlike the previous example.
Since these can be readily driven straight from the portable source with not just enough overall VOLUME but in fact more than enough it comes down to the little things.Better bass impact,more dynamic with busy music where instead of a single wall of sound you get actual note separation and with the good amps can follow the single instrument even when surrounded by many others.
This type of headphone does not need voltage gain for volume but output
DRIVE capability so really more impedance converter than it is amplifier.
In the end an amp allows matching need to device and whether it is voltage gain or current gain all about making a signal meant to drive a 10K and up load now able to drive a much lower impedance load without LOSING information !
Not about what an amp increases (volume) as much as limiting what is lost