For the purposes of bedding material, it makes no difference--people generally would choose what's most abundant, readily available, or cheapest. Dry leaves (preferably shredded to be more absorbant), wood chips, shavings, etc can all serve well also. You can use whatever is available as a carbon-rich, absorbant bedding--provided it is free of chemicals and non-toxic, obviously.
Straw is just the stems of the plant. No seeds or anything nutritious. All that's been removed to be used separately as feed. It's great for bedding, but not the most absorbent material.
Hay is the entire grass or grain, harvested with the seeds intact. those seeds and the fine stems of the grasses are what provides nutrition for the animals that eat it. It can be used for bedding, again not the most absorbent material.
I give my birds the hay that breaks off the bales and winds up on the floor. I'm not sure how much of the seeds or other good stuff they get from it, but they sure enjoy scratching around in a pile of it, looking for whatever good stuff is there.
I've been having pretty good luck with bermuda grass. Laid down deep, I don't smell any ammonia or poop. I do throw a few handfuls of scratch in now and then to get the girls to mix it up. It's been a few months now, and I just add more to keep it thick.
Different grains, depending on the maker. It's usually corn, wheat, oats, things like that. No added vitamins or added protein like a "feed" has. it's kind of like chicken candy or potato chips. it does have some nutrition, but it's not a complete feed. it 's a treat, or something to encourage them to "scratch' around in the bedding to entertain themselves and help the human by turning the bedding over.