This seems to be the general consensus, but 99% of these people are referring to a Les Paul. What if the neck was maple? Would a big maple neck still be fatter than a smaller maple neck, or would the tone just be "more maple"?
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A fatter neck doesnt mean a fatter tone at all

The material the neck it made of + your choice of pickups play a much bigger role in the shaping of tone.
Disagree. Tonewoods make your sound fatter. Dense mahogany guitars with thin necks can sound HUGE! Mahogany just happens to be the wood Les Pauls use.

In terms of the general maple question, I'm pretty sure that it has no bearing on the tone as the wood surrounding the pups is arguably the most important.
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Someone on here will argue that having a fatter neck will ensure fatter tone, depending on the wood being used for the neck.
If you want to find out, get a Norlin-era Les Paul Custom. Many of them had maple necks rather than mahogany.
Way too many factors come into play for a simple yes or no answer. In general, both the amount of wood and type of wood will make a difference. If not, then there would be no difference heard between maple, mahogany, rosewood, or balsa.