Frilly, straw comes from grain crops such as oats and wheats -- after the grain is harvested the stems become the "straw." Hay is grown from other plants, mostly alfalfa and Timothy up here. As Porkpal pointed out, the hay is cut before the plants go to seed because it's more nutritious.
I actually think the amount of seeds in hay or straw would depend mainly on how many weeds were growing in the field and not as much whether it's hay or straw... maybe Porkpal or someone else can give us an opinion on that thought.
