Straw is a form of bedding and has very little if any nutritional value.
Hay is feed for horses, cows, goats, sheep, llamas, alpacas, rabbits etc. There are several types of hay. Alfalfa hay is very high in protein and some consider it to be harder on the internal organs, kidneys in particular. Thirty years ago, alfalfa was considered premium feed but now, many consider it best suited for cows. In my area, orchard grass is a very popular type of hay. Timothy hay is also common and seems to be most expensive. Although I know people who feed it and swear it's the best hay, I don't agree. The reason is my mare, who was in a barn for two years that fed only timothy, she now turns her nose up at it. If there is nothing else to eat, she will pick at it but she does not finish it. I personally like a baled mix of orchard grass and alfalfa (ratio of 80/20 or 70/30).
There are also various cuttings of hay. This basically translates to how many times the hay field is mowed and bales formed during one growing season. The first cutting does tend to be coarser and some of the stalks may be more straw like. My geldings will eat up every bit of a first cutting orchard alfalfa mix but my mare will leave behind the coarser stalks. Since she is so picky, I have to buy second cutting for her which is finer and softer. My goats prefer it as well.
Second and third hay cuttings are a bit more expensive and is higher in protein.
First cutting is higher in fiber and is good hay if the animals aren't too picky. Depending on where a person lives geographically, there may be some differences to consider but what I've shared is based on my knowledge and experience from living in the Pacific Northwest.