A headshot is looseley defined as "a photograph of a person's face or head and shoulders."   It isn't to be confused with a loosely defined description of a portrait "a painting, drawing, photograph, or engraving of a person, especially one depicting only the face or head and shoulders",   which let's face it definition wise, they sound very similar....
My personal thoughts are a headshot is what you look like in the mirror at your best, with minimal fuss in the background and a resonably nuetrual look on your face. A portrait is more of an insight into who you are and who you want to be, or who you want to be right now. It doesn't have to be constrained in anyway. From composition, to backgrounds to post processing techniques.
Your professional networking profile image (LinkedIn/Corportare headshot)
Social media profiles
Creating your personal brand
To help represent you when you can’t be there in person
Dating agency profile
Anywhere you'd like to include a picture of yourself
Business cards with a photo of you for networking events