What is the difference between "rope" and "cable"?
Jonas, welcome to English Stackexchange.
I know you said the difference doesn't seem to be material, function or thickness, but it really comes down to material.
Cables are typically composed of metal, wire strands and rope is composed of natural fibres. I say 'typically' composed of as cable can be composed of natural fibres, but this is not typical. Rope can be comprised of metal, but is then referred to as 'wire rope'.
The usage of 'ropes' for your example of cable driven robots would not be suitable.