Hay is green and is fodder for animals. Straw is usually from harvest wheat and only consists of the stalk of the wheat. Hay can carry lots of pests and seeds. Straw carries less seed but typically isnt used for a weed barrier as much as its to retain moisture and keep dust down...
It is easier to pull wheat that other weeds ime. Just make sure soil is warm before you put down any weed barrier that is not a plastic mulch or ground cover, otherwise your soil stays cold for too long.
For increasing bio-mass the hay used as a mulch is going to more "more beneficial". I've often used straw mulch in the vegetable garden. Straw has a tendency to blow away in the wind. A light layer of stray creates a micro-climate good for seed germination. As mentioned straw has seed that will germinate in your garden. Alfalfa hay also makes a good mulch and incorporates nicely in the soil. If you're trying to prevent weeds from growing then the weed barrier would be "more beneficial"
BTW, weeds do exist! they are plants that aren't in the right spot.
Proper plants spacing reduces the need for mulch with the ideal being a full canopy of plants that creates a natural mulch cover
Yeah, what Barnyard said.
Pea Straw is the best mulch I've tried. When using as a weed barrier, don't break up the biscuits but lay them side by side. You use more this way but no weeds get through. Sometimes you'll get the odd pea sprout in the mulch and I just pick them and eat the leaves and stems.........delicious. It makes your soil zing and plants love it.