This is correct. Hay can be used as fodder for animals, whereas straw cannot. It can be used for many other purposes, however, like animal bedding, building material, decorating your suburban yard in fall...
Hay is cut from any sort of cereal plant early in it's lifecycle, specifically before the plant starts concentrating it's energy into the seeds. At this stage the plant stalk is sweeter (even to a human - give it a bite). After flowering, the plant is concentrating it's energy into the seeds. By the time it's fully done this (which takes a number of weeks), there is very little protein in the stalk, and it's far less palatable (or nutritious) to animals. The plant stalk is now essentially 'straw'.
Commercial hay can be mowed from a meadow (in Australia usually ryegrass) in which case it will have all sorts mixed in, or from crops intended for making good hay (in Australia usually oats or wheat). Commercial straw (which has a tiny market) is cut after the grain has been harvested from the top of the plant. In commercial broadacre cropping in poor soil areas (the bulk of Australia's grain areas) it's usually better economics to keep your crop residue including straw since the cost to replace the carbon would be higher that what you'd get for the straw after the cost of harvesting it.
Hay is generally cut from fallow fields in late summer/early fall. It contains sugars in the stalks as well as the seeds/grain of whatever grasses it is cut from. It is used to feed livestock year round and is the primary feed in winter. It is often supplemented by grain or feed.
Straw is basically an industrial byproduct and is the leftover stalk from grain harvesting. Usually produced from wheat after harvest. Used as bedding.
Hay is basically cut grass, straw is the part leftover from harvesting wheat and taking the seeds. Both are baled, but they're used for different things. Hay is food for any animals that eat grass like horses and cows, buy straw is not edible so it's used as bedding.
I remember being told even horses can't get nourishment from straw. But it has other uses. Mixed with mud it can make bricks, it can be burned for ash, it absorbs gross fluids, it can provide padding and insulation.
I know everyone that follows this link is guna be like:
"wtf, who posts an entire hour long comedy special in a thread about being uninformed about faming? You couldn't time stamp the relevant joke?!?!"
To which I will reply:
"I did time stamp the relevant part of the special. Jokes on you cuz it's a whole fucking hour long joke about not knowing anything about farming!"
Its so funny by itself but its better when you know it was all done in front of an audience chalk full of local farmers. I found it getting ready for a road trip with my 3 y/o niece in the car and was shocked at how laugh out loud funny it was while also being completely G rated. Dude doesn't even say the Simpson's swear words: shit, damn or crap.