After looking up examples and doing some research in dictionaries, the difference doesn't seem to be the material, function or thickness.
In particular I am interested in the usage of the word cable for cable driven robots. They exist in various sizes and materials. Typically the "cables"/"ropes" are made of metal or synthetic fibres. Could the "cables" be called "ropes" in this case?
The Oxford Dictionary Online defines cable in various ways but the most fundamental is
A thick rope of wire or hemp used for construction, mooring ships, and towing vehicles.
so the main difference is that a "cable" is a particular type of "rope", that is one that is large in diameter and, by another, nautical definition, around 200 yards long. The material from which it is made has no significance, a cable is a long, thick rope.
There are other definitions of "cable" mostly relating to electrical cables of some sort or another but these are derived from early electrical wiring's visual similarity to braided hemp, linen and steel ropes known as "cables" by the original definition.
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.
It is, of course, obvious that originally "rope" was made of vegetable fibres, and that wire rope and the wire cable emerged much later and after the word "cable" = a thick rope had appeared.
OED
rope, n.1
I. A stout cord, and related uses.
1.a. A length of thick strong cord, made by twisting together strands of hemp, sisal, Manila, cotton, nylon, wire, or other similar material, typically used for pulling a heavy load or for tying up a bulky object, and esp. forming the major part of the rigging of a sailing vessel or used to assist a climber.
And then we have
cable, n.
1.a. A strong thick rope, originally of hemp or other fibre, now also of strands of iron wire.
But this is the crux:
Originally a stout rope of any thickness, but now, in nautical use, a cable (of hemp, jute, etc.) is 10 inches in circumference and upwards; ropes of less thickness being called cablets or hawsers.
In other than nautical use (see 2), rope is commonly used when the material is hemp or fibre (as in the ‘rope’ by which a train is drawn up an incline), and cable when the material is wire.
1708 J. C. Compl. Collier 13 in T. Nourse Mistery of Husbandry Discover'd (ed. 3) A Cable of three Inches round and of good Stuff, will do better for Coal-work.
Look at the date, we are talking about rope - vegetable fibres twisted together
1842 Penny Cycl. XXIII. 336/2 The platform [of a suspension-bridge at the Isle of Bourbon] is suspended from four cables..and each cable consists of fifteen bundles of eighty wires each.
Wires...
The answer to what is the difference between rope and cable is that generally rope is made of twisted vegetable or artificial fibres or thread, whereas a cable is made of metal.
There is an overlap where thick ropes become cables in nautical terms and thin wire cables may become ropes.
This overlap does not exist if the cable is designed to conduct electricity or light, for, regardless of anything else, it is a cable.
If you are looking for a legal or scientific definition of the two that will hold for, and distinguish between, all cases, you have just been disappointed - it does not exist.
