Pros and cons to both.
Cotton: Does not hold heat as well but won’t chill a horse, either if the proper weights and fills are applied, more absorbent than nylon but some horses don’t care for the friction, ‘specially when sporting a fresh body clip, tends to grab fur and pull in the wrong direction. The fit of a cotton sheet should be more precise to avoid particular rubs, Cotton, depending on your wash system, does not launder or dry as easily as nylon. Has more of a wicking nature than nylon, holding more sweat/water thus can get heavy.
Nylon: Less friction, lighter weight, more form-fitting and easier to fit (doesn’t need to be AS precise as cotton) due to stretchy properties of nylon, warmer more than cool, dries faster than cotton but static is more intense.
Bottom line: Both can be found in just about any shape, size and colour : )
The material choices come down to what fits your horse best, what you are blanketing for/against and what s/he prefers. My TB mare will NOT wear cotton/flannel due to the friction against her clipped self, makes her grumpy, she also sweats up more readily in cotton. Nylon conforms better over her sharkfin withers (she is most regal and has been trying to tell me for years that Chiffon and Taffeta are HER preferreds). My gelding gets all snuggy and nestley in his cotton and it sets well, stays put around his entire football team shoulders and Mississippi chest. He is hard pressed to relinquish it when blanket changes take place. My 17 year old baby would wear burlap and tinfoil year-round and proudly.
This is what I have found over the years, can’t speak for anyone else. No right or wrong, tho, other than fit, and as I am sure you well know: one man’s wrong lead is another man’s countre-canter. What promotes one will shut down another. Good luck in your research, wishing you a warm and happy horse thru winter : )
Sure there's right and wrong but mostly there's just a whole lotta different.