I love this subject, though I can only address the tip of the iceberg in the space allowed here. I will stick to the tried and true laws of physics and spare you any personal opinions.
We just discussed how 45 style binding materials affect the sound of the guitar and the same principles apply to glues. If a material is resonant it will transfer vibration and if not it will resist vibration from passing through it.
In respect for the violin tradition we choose to isolate the top from the sides with a non-resonant material to direct string energy back into the top to pump more air (sound volume) out the sound hole. The alternative would be to connect the top to the sides with a resonant material to send the energy into the sides directing the volume to the player.
Here you can see how the difference in the glue’s hardness would allow the same manipulation. White poly vinyl resin never gets hard while hide glue dries like glass and yellow aliphatic resin sets in between the two in hardness. Each will transfer vibration at a different velocity and each will contribute to a different “tone” along the scale of bright/clear to warm/dark.
As an Example; Eric Skye’s jazz, classical finger-style is best represented from a guitar with a quick response and clear, articulate tone. We accomplish this, in part by attaching the top bracing and bridge with glass-like hot hide glue. We amplify that clarity by using stiff, bright Adirondack Spruce for the top bracing. Conversely; we use softer glue for the top to side adhesion to keep vibration within the top instead of allowing this energy (sound volume) to transfer into the sides.
Like many of our techniques for controlling the various parts of the guitar’s sound we will use several subtle manipulations that when combined produce the desired results. This is why we only offer hot hide glue when combined with Adirondack bracing and a brighter toned wood for the top. This ensures that we get a genuine difference in tone in which the use of hide glue plays a valuable part.
All the best, Richard