A vest is a garment with no sleeves. A vest that you wear under clothes is usually called an undershirt. Vests as part of a suit are usually called a waistcoat.
The baby in the picture is wearing a knitted vest.
A wetsuit is a special garment that people wear in the water. People usually wear wetsuits when they are scuba diving or surfing. Wetsuits are made of a special material that keeps you warm, and protects you from getting cut by rough stones or coral in the water. Wetsuits are also slightly buoyant, which means you float better if you are wearing a wetsuit.