Ground hay also allows the cow to eat more of the plant. With most feeds like alfalfa, the feed value or protein is in the leaves. Given the choice the cows would eat the top third of the alfalfa plant, where most of the leaves are, and leave the bottom two thirds. The stems of the hay also act to 'scour the rumen' which aids in digestion. By grinding the hay the tough stems are broken down in to chewable pieces for the cow and the good smelling sweet tasting leaves are combined with the stems so the cow is more inclined to eat all the feed.
Also, for trucking feeding hay any distance, grinding works better because you can get more bales worth of ground hay in a 53' walking floor van trailer than you can haul big round bales on a flatbed.
Another good point for grinding is that you can easily vary the ration. I had customers that would have a pile of ground alfalfa, a pile of ground wheat straw, and a pile of ground corn stover. Depending on the animal's feed needs, the ration could be varied quite a bit. Lots of farmers in our area are now using wet distillers grain, which is a byproduct of corn ethanol production. This is a high protein feed which can reduce the need for some alfalfa, allowing substitution for other cheaper roughage sources. Almost all the roughage ingredients need to be ground so that they can be properly mixed with a conventional feed wagon.
Plants like prairie hay, cane, millet, sourgum (we call it milo) can all be ground and have the same benefits when ground as alfalfa. In fact, with cane, the stems contain sugar which the cows like but they don't like chewing the tough outer stem to eat the core of the stem. A grinder breaks down the hard stem shell and the cow will eat much more of the cane than before.