oh lots of things really. dynamic range (difference between the loudest and quietest bits), seperation (how well it can distinguish between left and right signals), imaging (how well it can 'place' a particular sound or instrument in the sound field), power handling with difficult loads (low impedance headphones for example), the flatness of the response (how even it is across the range of the audio spectrum), the amount of distortion, the amoutn of background noise.....
it goes on and on basically lol
headphone amps will drive a pair of 'phones much cleaner and with a much better presentation, in a nut shell. them may not necessarily go louder unless your headphones arent very forgiving of the amp. but expect tighter bass, a smoother midrange and a much cleaner top end