In journalism, the common questions to be answered about a subject one is writing about are: Who, When, What, Where and So What.
That should answer the question. Who and what can be seen as subsets of that basic rule of thumb.
Of course, one could answer the question other ways, to wit:
"What are you?"
I am a proofreader. I am a beer drinker. I am an ankle biter.
"Who are you"
I am my father's son. I am a stranger in these parts. I am the cat's pajamas.
The philosophical answer, however, would be: "What" asks the question in terms of an object. And "who" asks the question in terms of the subject. Something we hear less and less about in these days of everything being about things being in one's DNA. (How does one locate talent in genes, anyway?)