Straw ... Hay is to prone to molding quickly. Then you end up with respiratory issues with your chickens.
I don't much like either. Straw harbors parasites and as the previous poster said hay does mold. Neither is very absorbent. I have used hay some in the past, you have to clean it out very often. I prefer pine shavings in the coop. I just scatter a few on the floor, but fill the next boxes good. Every few days I have to add a handful to the nest boxes and before you know it the floor has a good layer. I just go in and scoop out dirty spots and mix the shavings up. It only has to be raked out and scooped every couple of months if you do regular house keeping. I don't use either of these in brooders. I use corn cob bedding, but most folks here use pine shavings.