I can read the dictionary definitions myself, of course:
Someone who is strong-minded is determined and unwilling to change their opinions and beliefs:
You'll have to be strong-minded if you're going to push the changes through.
very determined to do what you want without listening to others:
She was a headstrong child, always getting into trouble.
So from these, headstrong has a more negative connotation, more or less meaning stubborn in inappropriate circumstances, whereas strong-minded might mean tenacity in general, perhaps with positive tint. Is this a good summary of the distinction, or am I inferring too much from these definitions and the terms are more liberally interchangeable in common discourse?
N.B. If google is not mistaken, 'strong-minded' sees less usage than 'headstrong' (or 'strong-willed' nowadays), perhaps because 'strong-minded' is more ambiguous.
But simply looking at n-grams doesn't help with context.