Quality of hay always depends on when it is cut, how it is dried and when it's baled. It's lower quality if it's been rained on previously before being baled, and if it is baled too soon, causing mold to form on the inside of the bales. It also depends on what species of grasses or legumes that are in the hay. You will get different quality hay between first and second cutting. For us, first cutting is the most grassy (or stemmy), and the second cut has more legume than grass in it.
Typically the first cutting is lower quality because the hay isn't cut until the grass just starts to seed out, and also because the grasses tend to grow much faster than alfalfa during spring time. Hay needs to be cut not only when the seed heads of the grasses come out, but also when the alfalfa starts to bloom. Cut too early and you either eventually take out the alfalfa or you have a higher risk of having to deal with bloat in your cattle, because nothing's more bloat-prone than pre-bloom alfalfa, especially if your hay fields have a >/=50% alfalfa stand. It's a different story if we are referring to straight grass hay or hay that only has 20% alfalfa/clover in it or hay that contains legumes, like sanfoin, cicer milkvetch and birds-foot trefoil that are bloat-resistant. Our neighbor runs a dairy operation and he hays his timothy-orchardgrass stand that has 20 to 30% alfalfa before it starts to seed out in the spring. And he cuts it when it is over 12 inches in height, which is also optimum time to graze for lactating cows and young cattle.
On the other hand, second-cut hay is higher quality because the legumes have taken over the stand and most of the grasses are not at the seed-head stage. But again, when you cut (among other things) is important in determining quality of the hay you get.