Nylon rope is stronger than manila rope. That is a fact, and for applications where maximum strength at minimum weight is the requirement -- climbing, rigging, marine use in saltwater -- nylon or polyester rope is the correct choice. For most other applications, the case for natural fiber is practical rather than sentimental.
Manila rope is made from the leaf stalks of the abaca plant, a relative of the banana native to the Philippines. The fibers run along the full length of the stalk and produce a strong, flexible rope when twisted together. Manila has been the working rope of farms, boats, and industry since the 19th century for three reasons: it is strong enough for virtually all non-climbing applications, it grips when knotted in a way that synthetic rope does not, and it is biodegradable.
The knotting behavior is the practical advantage most often overlooked. A cleat hitch in manila rope bites in and holds without slipping. The same hitch in nylon rope can slip under load because the smooth surface does not grip itself. A bowline in manila maintains its loop shape and can be untied after loading. A bowline in nylon under significant load can compress so tightly it requires cutting to remove. The texture of natural fiber is a feature, not a defect.
Sisal is a coarser, cheaper fiber from the agave plant, used primarily for baling, bundling, and garden applications where the highest strength is not required. It is the right material for tying tomato plants to stakes, bundling produce at harvest, and any application where the rope will be discarded rather than stored. Sisal is biodegradable; it returns to the soil without leaving synthetic residue.
Jute is the softest of the common natural fibers -- the material used in burlap and twine. It is used for decorative applications, light bundling, and any situation where softness to the touch matters. It is not a working rope in the sense that manila is, but it is the right material for its category.
The case against synthetic rope for farm and garden use is partly environmental and partly practical. Polypropylene rope is petroleum-derived, non-biodegradable, and floats in water rather than sinking -- which creates hazards for wildlife and is simply wrong in most farm and garden contexts. Manila and sisal biodegrade within a season when left in wet soil. The rope that gets buried with the bean plants in October is gone by spring.