For years, cotton has enjoyed a reputation as the desirable material for T-shirts, primarily due to its soft feel and natural breathability. Polyester suffers for the opposite reasons -- synthetic materials tend be hotter and scratchier than their cotton counterparts; however, given the variations in application and how the material is woven, there's ample reason to have both in your closet.
Cotton is a little cooler in the warmer months when T-shirts are typically worn as an outer layer. However, because cotton is very absorbent, sweat has a tendency to hang around in the weave of the shirt. Polyester shirts have inherent moisture-wicking properties, most notably leveraged in compression athletic gear, which is primarily made of synthetic materials -- polyester among them -- to keep sweat away from skin and allow you to continue to effectively cool yourself.
Cotton's softness and overall comfort comes at the cost of decreased durability, at least when compared to a polyester alternative. This is especially noticeable when comparing the shrink rates between cotton and polyester T-shirts. Drying cotton tees in a hot dryer is sure to make it fit much more snug than when you threw it in. Conversely, polyester is virtually immune to shrinkage when exposed to conventional washer and dryer temperatures. Cotton also abrades more easily than polyester, making it look more worn faster.
Nearly any cotton shirt fresh from the dryer is in need of ironing to one degree or another. Cotton tees are more prone to wrinkling and deforming, requiring more hands-on care on a day-to-day basis to keep them looking fresh and crisp. Conversely, polyester is quite resistant to wrinkles -- straight out of the dryer the piece will look comparatively smooth and flat.
Sports apparel and activewear for men and women often feature 50/50 cotton and polyester blends, taking advantage of the breathability of cotton and the durability of polyester. This also makes this type of T-shirt easier to clean as it becomes soiled from activity and perspiration, since polyester is naturally resistant to stains. These fabric blends also make wash-and-wear more of a practical option, allowing wearers to use a fluff cycle in the dryer over an ironing session.