Does anybody know if a headphone audio amplifier actually helps improve the quality of the music? Not interested in volume, just quality. Thanks.
A chain is only as strong as its weakest link. And it's particularly true for audio.
If the recording, or its encoding, or the DAC, or the headphone, is of bad quality, it won't make a difference to have a good dedicated headphone amp. Or it could even be worse since you may be able to hear distorsions more clearly.
Conversely, if the recording is very good, and you use a lossless CD-quality (or higher) audio encoding and a good headphone, using a built-in computer sound card or an entry level smartphone could really make it dull.
For an audiophile-only use, a dedicated headphone amplifier is a must indeed. For instance:
For a more versatile use, such as computer-aided music, an good audio interface could provide reliable audio listening experience too. For instance: