I reckon the amount of seed in straw will vary a bunch depending on the farmers practices and how weed free his fields are. At any rate it is apt to have some seed from the crop in it too.
"Hay", can be a lot of different things. Basically it is green plants of grass, clover or alfalfa, cut when green and dried for stock feed. If it is a good thick stand of alfalfa cut before it is in full bloom, like most farmers like it, there won't be much seed in it because the alfalfa crowds out the competition, and if the farmer cultivated it in the spring , he killed most of the weeds.
Any way either will give you good organic matter. Use what is available and inexpensive. As a rule, in these parts, straw is much less costly than hay.
No matter where you get organic matter, if you truck it onto your fields, you may be asking for new weeds.