The only difference between straw and hay is the time that they are harvested.
Hay will be greener. It is green growies cut and dried while green. Hay is usually used as animal food - like feeding
cattle in winter.
Straw is the same stuff allowed to go dormant. Grasses will often turn yellow. Most straw comes from grasses that went dormant complete with a big seed heat. Grass seed is called "grain". The grain will be nipped off the top of the seed stalk by a combine leaving the yellow leaves and stalks about two feet tall. What is remaining will be cut and baled. Straw is usually used for bedding for animals. Sometimes the animals will nibble at it.
Both hay and straw can contain seeds.
Both hay and straw can contain weeds.
Both hay and straw can contain weed seeds.
Hay makes a great mulch for plants that like a lot of nitrogen. Straw makes a great mulch for plants that prefer less nitrogen. "Strawberries" get their name from doing so well when mulched with straw.
Most hay and straw contain persistent herbicides. If you use hay or straw with persistent herbicides as a mulch, your growies will be stunted or die. Except for grasses (the persistent herbicides are typically broadleaf herbicides that are designed to kill anything but grass).