"Half-blood" in this book's title is a noun adjunct,
formed from the slang(?) term in the Harry Potter universe for a
witch or wizard who has one non-magical parent. It is derived from
the noun "half-blood."
"Half-Blooded" is a participle adjective (just like
"cold-blooded"), and also derived from this in-universe slang, now in
verb form. Being "half-blooded" means "being endowed with half-magical, half-muggle lineage".
Grammatically, both function as adjectives and convey the same meaning. In terms of nuance, the difference is more subtle.
To my ear, "The Half-Blood Prince" sounds like a specific title or honorific given to a particular person, i.e. this prince's identity is entirely defined by his half-blood status, so "Half-Blood Prince" is a proper noun.
However, "The Half-Blooded Prince" seems more like "the prince who happens to be a half-blood," but that detail is of lesser importance and does not define him wholly.
One might call any prince who is a half-blood a "half-blooded prince," but there is only one Half-Blood Prince.