Well, it depends where you live. There is a really big difference in the plains states. Cow hay can be poor quality like hay that is not cured well or baled too soon after rain and it is dark and dusty from mold. Some of these molds are toxic to horses. Cattle can digest hay that is black, dusty, has gray and white mold in it --- stuff that would kill a horse.
We keep cattle and a few Bison just to feed bottom bales and spoiled hay to so we don't just waste it. Then, we have the cattle or Bison to work with our horses.
There is also a huge quantity of hay baled for cows that is Johnson Grass, Hay Grazer or other coarse grasses in the Sudan family. You can also buy baled wheat, straw, corn stalks, etc. These are definitely not suitable for horses.
If you have broodmares or young horses, you do not want to feed Fescue Hay. It is the cause of more breeding, foaling or foal mortality problems than anything else in the central and eastern United States. Just Google 'Fescue Toxcicity in Horses'.
If you feed alfalfa, most of it is tested and is sold by its RFV value. That stands for 'Relative Feed Value'. Good quality grass hay (like good Brome Grass) is the standard at 100 RFV. Dairy quality alfalfa that is bright green and very leafy runs 185 to 200 RFV. It will usually run about 25% protein. Most horse breeders and trainers around here stay with hay that is tested at 140 to 160 RFV if they are going to feed alfalfa.
If you want to see how hay is listed in the US, go to Internet Hay Exchange - Hay For Sale
. I have bought hay from it when we had a drought in Oklahoma and I found semi-loads on this site. You can see that there are many kinds of hay for sale -- not all are suitable for horses.
I hope this clears up some of the misconceptions about hay -- at least in the US.