Cotton and Lycra swimsuits feel much like a thick, tight tee shirt. The Lycra makes it fit snugly and helps it keep its shape over a long period. Because the cotton 'breathes', it tends to be cooler in hot, humid conditions. On the down side, if not taken care of, it will fade easily. To guard against fading, and stretching out, wash your suit in a chlorine removing cleaner after each wearing.
Nylon and Lycra suits shed water quickly, shortening drying time. They glide through the water with a minimum of friction, making them the suits of choice for racers. The Lycra is likely to catch on the cement in and around pools giving a frayed and pilled look on the buttocks and stomach. Nylon and Lycra suits also fades, but washing in a chlorine removing cleaner will slow that down considerably.
Neoprene is used in conjunction with Nylon and Lycra by some companies. This provides some benefits beyond just looks. Neoprene keeps its shape for a long time, often longer than its companion fabrics. It also doesn't "give" easily, so in strategic places it can act like a girdle, enhancing your figure. Neoprene has to be 'constructed' like a wetsuit - darts or princess seams are the only way to accommodate a large bust.
100% Nylon suits have little 'give'. A ladies tank suit made before the advent of Lycra was very likely Nylon, as were men's racing suits. On the plus side, they are nearly indestructible; Nylon doesn't break down in chlorine nearly as quickly as Lycra.
100% Polyester suits have stretch and are also very durable. These are good suits for die-hard aquatic fitness people.