I can tell you there are differences to the sound of my two Angelii that are greater than I would expect from the minor variance in their construction...same dimensions, frets/bridge/saddle/nut, tuners top and neck...different back/sides and FB (With Truss-BRW/Ebony, Without-Cocobolo/RW). I expected a little difference in the frequency response between the two...but the non adjustable rod guitar is much "woodier" sounding, considerably louder and sustains easily twice as long...longer than any other acoustic, HB or SB electric I have compared it to...except one...my PS Santana which also has a graphite reinforced neck..
You know, this is a really interesting post, and moreover, I have found the same to be true with graphite rods.
Both of my PRS acoustics have had them, and I've played others that didn't. I much preferred the graphite-equipped acoustics. I also think that people love prewar Martins for their woody tone, and they don't have adjustable truss rods - actually, if memory serves, they don't have truss rods at all, though I think my early 70s models had a T-brace of some kind in the neck. Hard to remember - they certainly weren't adjustable.
This is also going to sound a little weird given my love for the tone of wood, but the best sounding bass I ever owned was a Modulus Graphite Quantum IV with a carbon fiber 35" scale neck. Even the fingerboard was a bakelite kind of material. That bass had tone, and I mean, very good tone. Though the body was maple/alder.
I got it because it sounded great, but it was also pretty cool that I could travel to overseas gigs with it on airplanes, or go to a different climate with it here in the US at any time of year, and it never needed adjustment.
The same is true of the Tonares with the carbon rods. The necks do not move. There's never an adjustment needed, they are always set up perfectly, and they always play exactly the same.
Then, too, in my experience, a well set up guitar is better sounding than a poorly set up guitar, so there's another advantage of a neck that doesn't move with changes in ambient temperature and humidity.
But woodier? Yes. Definitely. Something to think about.