Which type of vertical window shades are the most durable?
I have vinyl in my new (to me) home (the blinds are probably as old as the house, about 30 years old), and my parents have had weighted cloth blind slats in their roughly 20-year-old home. Both have generally stood up pretty well, but depending on exactly what you expect them to withstand, they have different strengths and weaknesses.

Cloth:


Generally more flexible, so it won't crack or split when folded.
Better tensile strength; resists tearing.
Machine-washable (sometimes; often they'll wrinkle easily, but can be ironed out).
Cloth slats use metal weights at the bottom to keep them straight. These things cause all kinds of headaches, mostly by falling out or tearing the stitching of the pocket that holds them.
Cloth slats are almost always flat; even curving the ends generally does not cause a curve along the entire slat. This increases the gap between slats when the blinds are fully closed.
Cloth slats will attract pet hair; if you have cats or long-haired dogs, you will be forever trying to get the hair back off the bottom of each slat.
Cloth slats also attract pet claws, and will show damage from being scratched very quickly.
Cloth absorbs liquid, so spills will stain them.
UV tolerance depends on the exact material used. Any way you slice it, off-white colors are generally best as UV-bleaching won't change the color that much.
Most cloth slats are somewhat translucent, so light will get through even with the blinds closed.


Vinyl:


The slat itself is heavier, so there usually isn't a need to weight each slat.
More rigid form than cloth allows for the slightly-curved design that blocks light best.
Generally spill-tolerant (vinyl like most plastics is partial to red dyes, so if your kid spills cherry Kool-Aid on the blinds, get after that pretty quickly)
Doesn't attract pet hair, and claws don't make much impression.
Most vinyl is completely opaque, so vinyl slats will block virtually all light that would shine through.
More prone to shearing (tearing/splitting), especially with age.
Crease marks (from where the plastic was bent beyond the elastic limit) are permanent.
Hand-wash only with a damp rag; you can't just take them down and throw them in a clothes washer.
Will show puncture-type damage like chew marks more clearly than cloth.
Heat can more easily warp them, especially if they're not left hanging completely vertically (e.g. gathered behind a tie rod)
Generally a little more prone to sun-fading than most cloth.