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@moth
When you apply tension to the inner of a Bowden cable, you apply equal compression to the outer. Brake outers are formed from a single coil of square edged wire. They withstand compression by squashing the coils together. There can be a bit of give before the coils touch, especially on newly installed cables that aren't yet used to the path they're following.
Gear outers have many wires in a shallow (almost longitudinal) helix. These strands take the compression along their length and are only restrained from buckling sideways and bursting out of the cable by the plastic outer coating.
(In contrast, you could strip the coating off a brake outer and it would rust but still work.)
When you bend a brake outer you open up gaps between the coils on the outside of the curve, but on the inside of the curve the coils can't get closer together once they're touching. This makes the path along the middle slightly longer, effectively shortening the inner. These small changes in length with different cable bend are tolerable for brakes but bad for indexing.
When a gear cable is bent, (thanks to the helix) each strand of the outer will lie partly on the inside of the curve and partly on the outside, and there is enough give in the system to allow the slack on the inside to cancel out the lengthening around the outside of the curve, so the path along the middle of the cable remains about the same.
Gear cable style brake outers do exist, but they have an extra wrapped or woven layer over the longitudinal strands to stop them bursting out. I think they can be better all round, but are more complicated to make.
Apart from ends and thickness, i don't think there's any difference between brake and gear inners - the ones i've just looked at both have the same 1,6,12 concentric winding pattern.