Interesting question. Interesting enough that I'm curious, and so have spent a bit of time running numbers. You don't provide any specific values, so I've run a few of the more common or popular wire thicknesses... so we have a broader view of the possibilities.
The short answer, if resistance remains unchanged, is... it depends. ;-)
I'm using "Steam Engine" to calculate the numbers, the link to this coil calculator can be found as my sig line. So we don't go to extremes with wraps at either end of the wire gauge spectrum... In other words, so much wire that it won't fit into anything, or so little that there's not enough to wrap less than 3
coils.
I'll be using 1.Ω as a constant and I'm not going to go to extremes on leg length... more than double is sufficient to see a trend.
26ga wire 92.6mm long - 15/14 wrap - 1.5mm ID - 7mm leg =
7.6% power loss.
26ga wire 92.6mm long - 15/14 wrap - 1.5mm ID - 3mm leg =
3.2% power loss.
28ga wire 57.8mm long - 9/8 wrap - 1.5mm ID - 7mm leg = a
12.1% power loss.
28ga wire 57.8mm long - 10/9 wrap - 1.5mm ID - 3mm leg = a
5.2% power loss.
30ga wire 36.5mm long - 6/5 wrap - 1.5mm ID - 7mm leg = a
19.2% power loss.
30ga wire 36.5mm long - 7/6 wrap - 1.5mm ID - 3mm leg = a
8.2% power loss.
32ga wire 23.3mm long - 4/3 wrap - 1.5mm ID - 7mm leg = a
30.1% power loss.
32ga wire 23.3mm long - 4/3 wrap - 1.5mm ID - 3mm leg = a
12.9% power loss.
So... we can conclude several things from producing these examples.
To first answer your specific question -
For a given resistance which thickness wire performs better - thicker wire with longer legs, or thinner wire with shorter legs?
It's pretty close when there is only a few gauge separations. Slightly further apart when there are several gauge thicknesses in separation, and only in extreme do we have a substantial power loss (26ga/7.6%~32/12.9%)
For all practical purposes, it may not be that relevant, when resistance is the common denominator.... because, there is a wrap to wire length ratio that we may choose as most effective - within the given build platform physics we have to work with.
Extreme spread has more interesting information. With 26 and 32 gauge wire, the "rounded" wrap value is the same, ("exact" wrap, which I've not shown is slightly different to compensate for the added leg length) the leg and power loss only change. This is another possible indicator that, at extremes, either wire may be a poor choice for the desired resistance.
Potential power loss increases as gauge becomes thinner. This might change if I adjusted the resistance up or down, but might put either of our extremes out the running... at upper and lower resistance values.
I'll be putting this information in my blog so I don't lose it, and so I can run some additional formulas I've an interest in.
(If I've made any errors in math... someone please bring it to my attention)
Take it for what it is, do with it what you will. Cheers.
