Today most people probably don't know the difference between hay and straw. We've become such an urban/suburban society that basic farm knowledge is rare. And yet, a century-plus ago, supposedly a lot of new army recruits were so uneducated they didn't even know the difference between left and right, and hence had trouble marching when they were told (for example) to put their left food forward. The story goes that therefore the sergeants ended up tying bundles of hay to one foot and straw to the other, because the country bumpkins knew that difference at least.
Urban legend? Mocking story? Hard to say. But there is an old army chant (to be used when marching in step), "Hay foot, straw foot, belly full of bean soup, left! left!" etc.
Okay, medieval people knew the difference between hay and straw. Hay is basically tall grass, complete with seed heads. It is used for animal feed and is quite nourishing. Scything hay was one of the agricultural activities associated with the month of June, as seen in the mosaic below, from the monastery of Tournus. (Note the word Iunius, meaning June, on the left. Recall that there was no distinction between U and V in the Middle Ages.)
Hay could be grown on fields that were too steep or rocky for most crops, because there was no plowing involved, just let the wild grasses grow. Modern haying, which is done with machines rather than a person standing there with a scythe, requires a relatively smooth surface. Then as now, there was always a concern of unwanted weeds getting mixed in, because the whole purpose was to produce food good for animals to eat.
Interestingly, poppies, which are now treated as Good and taken as a symbol of veterans, were considered what we would call a Noxious Weed because they'd get into the hay field and grow up all prickly, ready to hurt an animal's mouth when they ate the hay. Poppies got their positive spin from a poem about the World War I battlefields, where in the years after the fighting, grass and wild weeds like poppies spread over the fields, "In Flanders fields, the poppies grow..." Medieval people still hated poppies.
Straw on the other hand is the stalks left after grain (the seedheads) is taken off for human consumption. The whole plant, stalk and all, would be harvested, then the grain separated out, leaving the stalks. The stalks would have very minimal nutritional value, unlike hay. Cows can digest cellulose, but even for them straw is pretty minimal as a food.
But that didn't mean straw was useless. Even now, straw is routinely spread in stables. It provides something for animals to stand on other than the bare floor, and it absorbs wastes, meaning that you can muck out a stable more easily by getting out the used straw than trying to clean a bare floor. Straw is also good for wiping things down, from a sweaty horse to a new-born calf.
Wheat straw was especially valued. Medieval wheat was tall, 6 or 8 feet, unlike modern wheat, which is only half as tall. Modern farmers don't want all that stalk, but wheat straw in the Middle Ages was used for thatching roofs, for weaving into baskets, and similar projects.
For more on medieval agriculture, see my new ebook, Positively Medieval: Life and Society in the Middle Ages. Also available in paperback.