I can't offer a medical opinion either, and I don't believe they are the same thing, but I can tell you that the ONLY time I have ever seen, heard, or read the word hyperthermia was in a first-aid class. People use the term heat stroke, but I suspect many of them misuse the term according to strict medical definitions. You will also come across the term heat exhaustion.
Hyperthermia is one of those medical terms that should exist but, actually, doesn't. Grammar Geek has it right when she says that the term has been misused. So much so, that it is creeping into some online medical dictionaries - but not all, and certainly not into those sanctioned by professional associations. The reason is simple - there is no consensual definition for the word.