It affects appearance and durability, but that's about it.
Of course, the unspoken second premise is that most musicians hear with their eyes.
Some do.
OTOH, I stripped and refinished a polyurethane-dipped Stratocaster with a thin coat of nitro, little filler, and the tonal differences, for better or worse, were significant enough that others noticed.
Also, IME subtle tonal differences using different finishes (and components) become less apparent the more gain and effects are used; Subtle differences in dynamic playing response become less apparent the more compression is applied.
I mention both because I've seen ostensible "proof that there's no difference" videos on YouTube where both gain and compression dilute the test data, leaving me to wonder "how could you tell?".