Both sound impolite if you are actually speaking to someone about their background.

Well, "half-blood" is a pretty obvious racial slur. "Mixed blood" is not such a great term either, but at least it doesn't imply the person is missing something (being half). I wonder: will we ever overlook race? We all have red blood.

The correct term is "mixed race".

mixed race

I think "mixed blood" and "half blood" have different meanings. "Mixed blood" refer to people from mixed ancestry. While "half blood" means the relationship that is shared by persons having only one parent in common, like "half-brother". I think you meant "mixed blood" in your question above.