This is how I remember it.
Straw is bedding, Hay is feed.
Either can be moldy depending on how/when it's cut and baled,
and how/where it's stored.
Straw is the stalk of any grain plant, wheat, oats, etc.,
left behind after grain is harvested...can be grain free, or not.
More likely to have been sprayed
and be GMO.
Hay can be a variety of grasses grown for animal feed,
can be real weedy, or not.
Probably less likely to have been sprayed or be GMO,
but not really sure about that.
Wish I had the experience to look at a bale and ID the plants and quality.
I throw down some thin layers of straw when run gets slippery.
Thin layers as I shovel parts of my uncovered run for access for both me and the birds. I keep a bale of hay too, toss out a flake here and there in the coop or in the dry 'under-run', more for activity 'enrichment' than nutritional value.
I have enough room to store a couple bales, to keep them from spreading all over my storage area I made 'bale bags' to keep them contained after cutting strings.