Everything matters. There's even a hierarchy of "matters" in my experiences:
1) Brain - if you can't HEAR the tone you want, you can't CREATE it.
2) Hands - if you can't EXECUTE what you are thinking, you can't get the tone out.
3) Strings - obvious: dead strings sound DEAD; fresh ones sound bright and snappy; between ones sound...well, some level of between...and you have to get the right variable to work with #1.
4) The instrument itself - with lots of sub-hierarchies: frets, fingerboard woods, neck woods, trussrods, graphite/steel/whatever reinforcement (if applicable), application of construction methods of the neck itself (quartersawn? flatsawn? multilaminate?), body wood/woods, hardware (brass bridge? aluminum? stainless steel?).
5) Electronics - can ONLY reproduce what came before. Important? Heck yeah! But, as a friend of mine always says, "you can't polish a turd", so if all the previous items aren't getting it done, you can't "fix" it here. You have to have a good solid set of tone variables PRIOR to this point, or nothing else matters.
6) Transfer of 1-5 from instrument/electronics to amplification - everything in the signal chain matters to some degree, depending on what it is/does. Cables. Tuners (true bypass? not?). Effects units can totally transform a tone into something unrecognizable, but it still needs to have a TONE to start with.
7) Amplification and equalization - Do we need to go into this rabbit hole?

Yeah, you can destroy a tone here, but you can't FIX a bad one.
8) Speakers - Again, a rabbit hole, many variables, but if any link in the chain is weak it won't matter here either.
How much can the wood change your tone? It depends on a few factors. I've said it over and over, I play hundreds of basses a year, all through the same cable, the same amp, the same hands, the same brain, in the same room, with the same pickups and preamp in 80%+ of them. Sometimes, it doesn't make much difference. Sometimes is makes a HUGE difference. It really depends on the wood/woods in question, both the species and the individual specimen.
Ash sounds like ASH - however, "ASH" is a variable tone, it will be within a "tone circle", but not always at a specific point. This is true of ever wood I've worked with...