What's the advantage of using cotton braid wires over normal wires?

The only real difference is aesthetic. While you might see some differences in durability, that would ultimately be down to the manufacturing quality, rather than the materials used.

I suppose the cloth-shrouded cables also have worse insulation, but that barely matters, as all the wires are individually insulated anyway, and these cables are not used for any voltages high enough for the shroud insulation to matter.