The number of turns determines the inductance. More turns = more inductance. More inductance means more voltage output and a lower resonant peak, less treble. That's rather theoretical though, the thinner wire might have different insulation thickness, or be wound looser onto the bobbin to decrease the risk of breaking the fine wire, so the wire thickness itself might not matter, but practical concerns might effect the final outcome. If a pickup has a ridiculously high DC resistance, beyond 15k, you can be certain it's made with 43 or 44 AWG wire. DiMarzio will use different gauges of wire for the two different bobbins in a lot of their humbuckers, so you'd have to measure the individual coils to see what they're doing. The pole pieces effect the inductance quite a bit also (due to permeability), AlNiCo 5 effects it the least, AlNiCo 2 or 3 increases it a bit more, while steel screws and slugs raise inductance by quite a lot.
The wire gauge doesn't really matter, because the voltage is too low to exceed the amperage limits anyway. Smaller wire makes for a smaller coil, which can effect the proportion of information the pickup sees from the string. More narrow window means more treble relative to bass, or more harmonic relative to fundamental.
As for the hookup lead wire, if the lead wire is shielded, like a PAF, the proximity of the lead and shield will add about 100pF capacitance, lowering the resonant peak, reducing treble response, the same way a longer guitar cable will. Strat type pickups with two separate lead wires have very little capacitance in contrast.
!
