Big difference. But first you should realize that heavier gauge wire is lower in resistance. So a 10k coil wound with something like 44 gauge wire will have a lot less turns on it than one would with 40 gauge. That would make the 44 gauge wire coil thinner and brighter sounding.
So the resistance of the pickup is not useful for gauging the tone, and is only of use when comparing two similar pickups wound with the same wire size. Then you would know that with two pickups wound with 42 gauge wire (the most common size), the one that reads higher is probably hotter, and wound sound darker.
Pickup makers generally wind to a number of turns of wire, and not the DC resistance.
To generalize the influence on the tone of wire sizes, thiner wire accentuates the mids. So if you wind something like a Jazz bass pickup with 8,000 turns of 42, and one with 8,000 turns of 43, the pickup wound with 43 gauge will have a lot more mids, and a "tight" tone. Conversely, if you wing 8,000 turns of 40 gauge, or heavier, it will have more highs and lows and less mids, and a rounder looser tone.
Sometimes thinner wire has to be used to fit more turns on a standard size bobbin. So a lot of hot wound pickups also tend to get mid heavy because of this.
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