The main difference between a head shot and a portrait, is that a head shot is a picture of a person's face. Its purpose is to identify someone, to see what their face looks like. A portrait on the other hand is something much more personal. A portrait is a likeness of a person to identify them, but it also tells you something about the person - or the personality of the person in the picture.
I have been in the photography business in the Kansas City, Missouri and Kansas areas for 25 years. I have seen the advent of digital photography, which has made photography accessible to everyone, but from my professional point of view, it still leaves something to be desired. Head shots can even be a smartphone selfie slapped on Facebook or LinkedIn. It does identify the person well enough, but does not tell you anything about the person. An expert photographer can take that to a much higher level. Of course a portrait takes more time and effort, and it involves the talent of an expert and reliable photographer. The person being photographed also has some input as well. It is a much more complicated and involved process, and involves more work after the photo is taken a well, and therefore does cost more, but is a worthwhile investment.
A headshot is usually just that. The person's face and perhaps shoulders, and that is about all. A portrait can take on a different look entirely. Often these are business portraits, but they have uses in any form of communication you might be involved in. I like to talk with the subject, and together we determine the message we want the portrait to convey. A knowledgeable photographer who you can trust will be reliable to help you create a portrait we can both be proud of.
In the end, we enjoy doing portraits because ultimately there is an artistic quality to each one, which is what makes photography fun. Consider the greater value of having a photograph that tells your story as it also shows your likeness.