Pete's article is great. Very balanced and I agree with everything except the conclusion.
In my scientifically unsupportable experience, materials may make a consistent difference, and I am mostly referring to necks and mouthpieces.
One point to specify: tiny or "negligible" differences can have a difference that is huge from the player's point of view.
The debatable points are that:
1- the so called "resonance" of the instrument makes the player play differently, so it has an effect on the tone production, not on the sound itself
2- the differences that I have in mind concern the harmonic spectrum in a very subtle way. Higher partials that require the use of often indefendeable terms like "projection" to be described. We often know what we mean, but we cannot formalize the concept.
The closest that I ever came to an experiment regards saxophone necks. I sometimes have access to a dozen necks in 65% or 85% copper brass (Sequoia standard necks). Sequoia does not particularly boast the qualities of the different alloys: saxes just come with two necks.
I proceed selecting three necks in the same material that sound pretty much the same. I must say that usually they are very consistent, and that the major differences within same-material necks are found in soprano pieces, where tiny dimension differences are more influential. On tenor they sound mostly the same.
When I have three of a kind, I compare them in a random way.
I found that in the Sequoia specific case 65%Cu tend to be more focused, the 85%Cu warmer.
At home I have a selection of necks that sometimes I use on guinea pigs I harvest in cafesaxophone. Interestingly the different resistance of different necks has a bigger effect on tone production, rather than actual tone, in the first blows. I also wasted some precious practice time matching mouthpieces and necks, and realized that bright and loud pieces seem to be more sensitive to the neck's material.
Mr. Sequoia swears there are no differences in the internal dimensions, but if someone wants to pay for someone actually checking, we can talk about it.