Wood does matter because the neck-body structure isn't ideally rigid, it has it's resonant frequencies, passes some vibrations and blocks other. It affects the sound as much as trampoline properties affect the way you jump on it, if you pardon this figurative comparison. Whether wood species is a decisive factor is a matter of some debate, as I have never seen an exhaustive and convincing experiment that would prove once and for all that wood matters or not, nor determine some general "tonal properties". Moreover, you can take the same piece and get different properties depending on the way you cut it.
Wood is chosen based on a lot of considerations: price, stability, looks, easiness of processing, and of course customer's perception. Fenders are made of alder ==> alder is cool, cheap Corts of agathis ==> agathis is bad.
First-hand experience: Squier Affinity with alder body sounds worse than Vintage Modified series with agathis body. Why? The answer is pickups and build quality. The accuracy with which neck pocket and heel were worked and put together. Solid cast bridge vs crappy pressed one (even though this is questionable, too).
