I see these as complimentary properties, or different sides of the same coin. A harder wood (e.g., maple) will result in a brighter tone than a softer wood (e.g., mahagony, rosewood). Maple and rosewood are common fretboard options on a slew of electric guitars (i.e., brighter or warmer). A harder material may reflect more sound content per its hardness, and in turn has less potential to vibrate than a softer wood. Wood vibrations (as airborne sound content that was not reflected) generally mask/destroy high range tones.
This video demonstrates the differences between the spruce and mahogany top options with the same model of dulcimer. This is an extremely simple instrument that differs only by one wooden piece, as the back portions are identical:
Acoustic instrument tone is both created and amplified through resonances and reflections. There is no debate that tone wood has a massive impact with acoustic instruments, and creating the best tone has been an art and science for hundreds of years. This is why Stradivarius violins sell for millions – they were the best of the era. In the 20th century, the science of sound came into its own and the production of quality tone is faster, better, and cheaper.
Tone considerations fully apply to the several cigar box instruments I built (even with my self-imposed spending target of $50 each), Body size matters. Type and thickness of wood matters. Sound hole size and shape matters. The height of the strings matters (i.e., as greater downward pressure transfers more sound energy to the resonant wood). This guy knows what he’s doing. I watched another build executed by a non-musician (which seems to have been taken down). The best comment was “what a waste of materials” – everything was done backwards and guaranteed to produce bad tone.
There are sharp debates among electric instrument builders about the value of tone wood. Yes, one can certainly hear and feel differences when playing a solid versus hollow electric guitar, as well as guitars made from different woods. However, electric processing adds a mountain of new and overlapping variables. Tone wood may have a minimal impact in the final product, or be overridden through later processing stages (e.g., tube amplifiers, effects pedals).
Headphones, as electric devices that require electric amplification, share many characteristics with electric guitars. Given the central role of electric drivers in sound production, one can use pure wood, a wood-plastic hybrid (stabilized wood), a non-wood composite, metal, or plastic to produce similar results. But audiophile-land is about the small differences. So YMMV and enjoy.