Is it hay fever or seasonal allergy worse?

Both the same: These terms are use for essentially the same process. I don't know many people who hang around "hay" to get their hay fever, although there can be a lot of moldy or allergenic stuff in hay. The term evolved to mean the same thing as seasonal allergy.

Same: Hay fever is the term for fall season allergy and rose fever is spring seasonal allergy. They are the same thing.