Hay is a bulky plant material typically used for feeding horses, cows and other livestock while straw refers to the stalk of a plant that has been harvested and had its grain or seed removed. The basic difference between the two is that hay has nutritional value while straw is less expensive but not nutritionally usable.
The primary reason for growing hay is for its nutritional benefit to livestock. The plant is grown and harvested at the exact right time to be most nutritious. Hay contains high levels of fiber as well as other nutrients that nourish animals and keep them healthy and thriving. All hay falls into two categories: legumes and grasses. Legume hays include clover and alfalfa while grass hays include oat, rye, timothy and orchard. The different types of hay come from harvesting the plant at different times in its life cycle.
Straw is not deliberately grown for any reason. It is the by-product that remains after viable crops have been harvested. While straw contains little nutritional value, it is useful in other applications. The stalks provide a dry, sturdy source of bedding for barn-dwelling animals and is unlikely to mold like hay does. It is also much less expensive than hay and is used in many craft applications, such as in the making of baskets and hats.