Hay I understand is a first cutting and contains all the seed heads, or 'grains' and that is nutritious for livestock.
The straw is a second cutting, of the stems part after some regrowth. (?) and does not contain matured seed heads. Therefore if you use hay, you will get lots of sprouts (weeds to your garden) and straw does not have that.
I use straw and I love it. I primarily use oat straw I buy at Lowes. I have Bermuda grass lawn and I HATE it really bad. Very invasive tough weed. Practically impossible to kill, I just try to control it at best. Even straw will have some viable seeds here and there. I would never purposely use anything that might maybe be contaminated with Bermuda seed. If you put the straw on in the fall, you may have to re-apply some in the late spring. usually spring and fall I lay some down. It looks a little 'raw' at first but in a couple weeks it settles in and looks nice. I just use mulch around the front of the house where I want a more formal look and the straw goes in my backyard.
