Ok, my previous post may have been a little confusing and ill written, let me try and elaborate on my point.
I don't mean to say that the type of wood used is unimportant to the sound, it is obviously a crucial factor. However, i mean to illustrate that not all instruments made with the same wood will sound the same.
Luthiers also work with different woods in different ways depending on their structural properties. For example, a builder would treat a cedar top differently to a mahogany top. They would be of different thickness and perhaps braced differently in order to get the best compromise in strength and sound from that particular material.
So the differences you would hear between the two instruments are not purely and simply a result of different woods having been used, but also of different methods of construction. You could not compare the instruments in this example as like for like. Of course the wood itself would play a strong role in the overall sound of the instrument, but what I am trying to highlight is that the methods of construction should not be downplayed as a factor.
To continue; Mahogany and cedar are two very different woods, one is a hardwood, the other a softwood, and therefore are often braced differently etc. Cedar is a delicate wood compared with the mahogany.
However, if a builder (lets call him Alfred) was to make two instruments to EXACTLY the same design, (as exact as is possible in the real world) one of say, koa, the other of mahogany. Sure, they would sound different from each other, and this would be largely due to the different woods.
I would argue though that if one were to compare these two instruments to a pair from another builder (lets call her Margaret), using different design and construction, the difference in this case would be greater still.
Alfred's mahogany uke would sound different to that of Margaret, even if it had been made from wood from the same tree, because they both make their instruments completely differently.
It is a black art, and its hard to define exactly how much each little factor will impact the sound of an instrument, but I think that people are far too caught up about what wood an instrument is made of when there are many other equally important factors that will influence the sound.