Sure, the different woods do not vibrate the same.
How different can the vibration in different woods even be? I would argue this is negligible, and would be at the end of the chain of things that affect tone.
And also: how much of the vibration of the body does even go back into the string to get picked up by the pickup? I’d again argue this is negligible.
I’m absolutely sure you can hear a difference, like @terb says. Different wood densities will transfer the resonance back to the strings with sometimes largely differing efficiencies. Hollow bodies sound dramatically different than solid bodies. Etc etc.
However, I am equally sure there are many other things that make bigger differences. Scale length, string type, string tension, age of strings, pickup type style and quality, characteristics of the electronics, etc. On and on.
So for me, the biggest factors for tonewood are more of weight (and looks and feel).
That’s my feeling too, Eric, @eric.kiser . . . 
It’s really a personal preference and there’s no ‘right’ or ‘wrong’ answer (like SO many other facets of playing bass). Fwiw, if the guitar is painted, I like a great shade of white in the body and in a matching headstock. If it’s unpainted, I like a nice “furniture” grade of wood with a great looking woodgrain . . .
Will it make me a better bass player? . . . NO
I will say, though, that a mahogany body has a warmer tone, while a maple body has a slightly brighter tone, based on my personal experience. Painted or unpainted, I like a white maple neck . . . 
I agree with the points that @howard makes above, too.
Just my 2 cents worth.
Cheers
Joe
This is a great point @Jazzbass19 but not for your context, but in @eric.kiser 's OP.
What about that thick paint?
What does that do to the tone?
Red - makes pickups hotter
Black - darker moody bass
Green - swamp like boogie
White - clean tone
So the argument is magnetic pickups in general will produce different sound based upon the variation/resonant from different types of wood and their density. The different types of magnet would yield more in your face differences than the woods. In addition, it would also influence the length/sustain of the note as magnet pull the string and lessen the oscillation of the strings.
The direct result of wood/ material types that influence sound and tone differences occur on fretless basses. The amount of “mawh” you get is quite noticeable between maple, rosewood, wenge, ebony, or synthetic materials like ebonel or richlite fingerboard. The more density the material the better it feels and sound.
I have 2 optical pickup basses which measure vibration by infrared light at the pickup with no magnetic interference. I don’t think that body tone wood have anymore influence on sound but the fingerboard materials do come in to play quite significantly.
So the fretless bit is certainly an exception for sure.
But back to the other piece, this I would agree with, in the way the pickup creates a tone due to stiffness/non-stiffness, vs. one wood coloring/sounding different just because it is different wood, its more about stiffness of materials. This makes sense.
The glorious tonewood debate XD
I want science. I don’t want “my superior magic ears can hear this, it’s so obvious, why can’t yours?” As that adds absolutely nothing to the conversation, tbh, other than another opinion.
I want the scientific method involved. That video was okay but could certainly use more data. Or someone with deep pockets and no vested interest in the outcome.
Identically shaped/built guitars made from different combinations of woods, a control guitar, etc. Preferably robotics that do all the playing identically and not a human. Spectral analysis, meters, etc. Let’s try scale lengths after that. Gibsons do sound different to me than say, Fenders or other guitars. Is it the scale length and the different tension? Where the pickups are placed? Is it just being made of mahogany vs alder, or basswood, or poplar, or whatever? Probably never happen because too many people “just know” and it sells guitars for them to be made of that special wood. And there’s no money in the research.
Someone with “silly money” needs to develop an interest in guitar tone 
Here’s an excerpt from an article I read comparing the Gibson SG to the Les Paul:
TONAL AND SOUND QUALITY
With the pickup, neck, and electronics remaining the same, it’s only the body structure that makes a difference in the tones of the two guitars.
As expected, the slender body of SG gives more depth to the mid-range chords. It also provides a little extra growl under gain. Although the difference is very little, but the crispiness and sharpness in its tone are enough to give SG an edge over the LP when it comes to rock music. The GS has more chime than an LP.
Although the harmonic feedback is higher in SG, it lacks in the depth and warmth of the Les Paul.
Crispness and sharpness in tone? Chime? I figured the LP would get that edge with its maple top (because I thought maple was ‘brighter’). Though it also has more mahogany as shown in the cutaway illustration. But I’m not a guitar builder, or any kind of expert on the subject. Could it be due to other factors? It even talks about an inherent “flimsiness” to the SG neck due to where it attaches to the body; is/could it be a a factor here? Did the author jump to wood being the main difference because that is already their bias? It was just the first article that came up on DuckDuckGo when I searched for ‘sg vs les paul tone’ and I haven’t gone too in-depth beyond that right now. I just knew Gibson makes mahogany guitars and those two came to my thoughts.
On acoustic instruments it immediately ‘makes sense’ to me; the sound itself is vibrating within an acoustic chamber to resonate and get amplified. But I don’t know the physics in the slightest so I’m not going to proclaim anything there, either, nor do I have enough experience with acoustic guitars to even begin to form an opinion. I’ve read articles on why a “Stradivarius” instrument sounds like it does, and it’s ranged from the specific makeup of the glue he used to the special wood that was from a bog or something that is no longer available because of how it was made in nature takes so long.
TL;DR - we humans are subject to bias all the time. There’s a lot more fact behind that than there is behind “tonewoods.” I think we need to be asking more questions and making less sweeping proclamations.
I can see this being a thing on a fretless where the strings are in direct contact with the fingerboard. My own experience with fretless involves two instruments that couldn’t be more different, though, unfortunately, so I can’t even offer up a dodgy opinion 
Ibanez GWB35 vs a Kala uBass lol…
Burn the witch!
It was so close to being something valid (from a scientific point of view)…
…sadly, there are many things that are different between recordings (recording mic seems to be in a dif position, human accuracy to replicate plucking strength and position, etc etc).
Lastly, if he wanted to ‘prove’ the similarities between the sound (or the impact wood has on guitar sound) it should be shown and compared using digital means.
Leaving something recorded with ‘some mic’ (there are no mic specs on the vod and, as stated above, the mic seems to be in different positions while recording the different variants), streamed into youtube without specifying the sample rates, then subject to people hearing those with speakers/headsets of multiple qualities makes the results of the test really subjective.
It was a really nice attempt tho.
Maybe, at some point, someone makes a serious attempt on showing the impact of each factor, in an objective and replicable way.
Why he used a mic and not a DAI is beyond me. This would have been much better.
I had a very well known sax instructor/player who waxed on about the tonal characteristics of ligatures (the thing that holds the reed to the mouthpiece) and how the design (holes, materials, etc) made the world of difference, and why he liked one vs the other. Many many people site these things. As an engineer/scientist by day, what I have found (without a lot of need for scientific method and hoopla, but rather looking at things in a very basic way, is that the difference he was hearin was HOW the reed was held and WHERE the reed was held. Move ligature up/down the reed - huge tonal differences can be noticed. Different designs lend themselves to positioning on the reed, thus the perceived differences being attributed to the wrong thing (simple engineering/physics things that musicians may or may not know or care about.
Add a dash of ‘i want my magical instrument and accessories to give me magical qualities from gods and unicorns’ and the myths continue.
I go back to something someone said to me a while ago…“Leo Fender was the cheapest cheap-ass around, and now people pay a lot of money for the ‘poorest quality’ vintage instruments”. Rewind further to saxophone history…why did manufacturers start silver plating instruments? - #1 - lacquer was not invented yet (came around for use in the 30s / @2 - many people have reactivity to raw brass and can start rotting the brass rather quickly via touch points / and finally - #3 - cause they could charge more for it (hooray marketeers!). Did it make the tone better? - depends on if you shelled out the extra moola and needeed to justify it.
Back to Mr. Air Guitar - this is a damn good experiment despite its shortcomings, a sound wave comparision could be used, but you don’t hear a sound wave, so who cares? What @Koldunya said - if you hear a difference, great, if you don’t, great.
The deeper research will never come. Every single acoustic research paper I have read (type of instrument aside) comes up very short in pure controlled experiment with scientific method (and I have read a ton, as this is a big interest to me). I agree there is no money in it so it is not done. In fact, there is negative money in it, if the results are ‘no difference’, so no one will ever do it. The ones that do are typically very short of science and get the result they want. Selmer has spent the last decade or so trying to ‘recreate’ the vintage Mark VI tenor sax sound (think Coltrane). They have sampled zillions of horns and zillions of iterations in design to ‘get back’ something those horns ‘have’. They have come up with nothing - why? - cause you can’t engineer marketing voodoo mystique (which can come from marketeers or users btw, in this case users). $12,000 gets you a good example of the MK VI voodoo these days, new Selmers are ~$8,000. If they could figure it out and get 50% more for the same product they would, but they can’t, because it doesn’t exist.
Tonewood = Fine Corinthian Leather = Dark Vintage Lacquer
“Cordoba . . . eet ees remarkable” . . .
Corinthian leather is soft and supple, and molds itself to your but cheeks far better than stiff Detroit leather which is more like cardboard.
And when Khan says buy a Cordoba you buy a Cordoba.
You hear what you want to hear. Whether its proven or disproven people will swear by what their experience is as we all do in so many other matters.
Its nothing I personally spend time with. There are so many other factors that make a big difference on shaping tone that deserve my attention.
Can’t say I can really follow all that’s been posted in this thread so far, but it is intriguing. So I found this series by Tim Sway, who plays stand up bass, and makes guitars and other things from reclaimed materials. He’s done a few comparisons in a unique way - by making the guitar (not bass) bodies from different woods, leaving a cut-out for the electronics and neck, which he has made separately using a piece of maple with rosewood neck.
In this video, he shows how he made the guitars, then samples each. 19:09 in length:
This one compares three woods, 3:51 in length:
This one compares ash, mahogany, and particleboard bodies, and 4:43 length:
There is a separate “reveal” video for the last video in the list above, which is linked at the end of it. He has some other videos on one or another channel, but these are the ones I viewed.
I ordered my Ibanez miKro from Guitar Center on Dec. 21, 2020, and it arrived on Feb. 22, 2021. The Ibanez inspection tag on it (for Spec, Set Up, and Final Electric) showed a date stamp of “20.9.21,” which I intially read as 20 September 2021, but obviously that can’t be right based on when I received it. Thus I suppose it is 2020, September 21.
The “Ibanez Wiki” site compiles data and puts it in a table, so I can see on its page for my GSRM20L model that in 2020, the body material is Poplar, the neck is 1-pc. Maple, and the fretboard is Jatoba. The link for Jatoba shows that the hardness of Jatoba (Janka score) is between Maple and Brazilian Rosewood, closer to Rosewood. The link for Poplar tells me that it:
If I’d bought a model made in 2015-2017, it would have come with a Mahogany body and Rosewood fretboard, but I didn’t, so no sense worrying about it. I’m inexperienced enough so that it really doesn’t matter while I’m learning to play. I do not have the ear yet (hope one day to develop it) to hear a difference in woods, as far as I can tell.
yeah, poplar has a softer attack than alder, but otherwise it sounds very similar. nothing wrong with poplar as a wood choice for a body. basswood is also very similar.
So is “Castillian Bronze” close enough to “rich Cordoban leather” for this digression? Because my first car was a 1971 Firebird Esprit in Castillian Bronze (image is not my former vehicle), with a 350 engine and a white vinyl roof. And as a young and foolish person, I survived driving it too fast!
Years later, I was no longer using it, it was in not-great condition, and taking up valuable garage space. So I sold it to my brother-in-law for $1, as he was going to restore it with his son (but son was not that interested). B-I-L rebuilt the engine, and I think did some other small improvements. A couple of years ago, he sold it to another guy, got a good price for it (no regrets from me), and that man planned on restoring the rest of it.
Here’s the one area where I don’t want any visual scientific evidence because this is the area that the different frequencies will show up between the 2 materials.
Same person plugging the same bass 2 different ways will yield different visual values.
Unlike most players I know I’ve been doing my best to play my basses in the most neutral position possible hoping to get the most default sound for each bass I own. I noticed that when it comes to wood on the body. What influence me the most is the shape and weight.
Dense heavy maple body like the one on my custom build Steve Harris p bass offered me satisfaction of the solid feel, same goes with most of my stingrays as they are heavy basses. The slap body of my stingray classic usually yields my willingness to dig in and get the extra juice out of it. I tend to play heavy basses one way and light basses another way.
Not to throw another curve ball into the mix but I oddly I can tell the difference between the different bass with the Piezo bridge pickups. I have several they do sound different with different wood I noticed that heavier body tends to give out better high end frequencies than the lighter ones which is usually darker by default.
Possibly because heavier wood, hardwoods for the most part, are denser.
here is a comparison of various woods hardness :
do what you want with this … but …
we said a few posts earlier that Alder and Poplar sound very similar, and their hardness happen to be very close. that said, Mahogany is harder but sounds way softer (less attack, less brightness, more warmth = more mids / more fatness). Ash is even harder but sounds way brighter than Mahogany yet warmer and with more sustain than Alder and Poplar.
what I’ve read on luthiers forum is that there is a concern regarding the internal damping of each wood. the more damping a wood has, the more the vibration will be attenuated when it travels through the wood. it changes the attack, sustain, and decay of each vibriting string.
and, obviously, every wood has its own resonnance frequency which will emphasize or lower various frequencies and harmonics produced by each vibriting string.
seems quite logical to me.
About 15-20 years ago, a friend of mine had a Danny Gatton Fender Custom Shop Telecaster. Maple neck with maple fretboard, light swamp ash body, Joe Barden pickups. At the same time I had a Telecaster with maple neck and maple fretboard, Warmoth Korina body, and the exact same Joe Barden pickups. Both guitars even had the same tuners and brass saddles.
We played those guitars into the same amps, same time, same place. both guitars sounded excellent, but it was absolutly impossible to say that both sounded the same.
