The short answer
Cotton is the safer choice if the overshirt will spend most of its life indoors. Nylon is more useful when light rain, quick drying, and bag-strap abrasion matter more than texture.
Cotton advantages
Cotton usually drapes better, feels calmer over tees, and reads less technical. It can be easier to style with denim, chinos, and knitwear, but it may hold moisture and lose structure if the fabric is too light.
Nylon advantages
Nylon handles weather and abrasion better. It can feel less natural against skin, so we look for a lining, soft collar, and enough ventilation before recommending it for long indoor wear.
Return risk
Overshirts vary heavily in shoulder width. If the merchant gives unclear measurements, choose the retailer with cleaner returns rather than the cheapest checkout.