for gardening mulch, hay is usually better, but sometimes straw is better

Just a note on straw—it can have mature seed heads as Paul described, usually because the crop was too low of yield or damaged by weather or disease right before harvest (not worth the money to run the combine harvester through the field) but it could have been harvested and then baled just the straw, in which case is partially shredded and has most of the seeds removed.

Hay can be made from annual grains, or from perennials—alfalfa, clover, any number of grasses.  If cut at the right time, there should be very few viable seeds, but it is possible to have a lot of weed seeds off the timing was off.

Any type can have herbicide and/or insecticide residue, so source carefully.