What makes some portrait photographers standout?

Maris said:

Yes, true, but it helps if the portrait subjects are high profile people, well known controversial or notorious people, politicians, and applauded figures from the entertainment and fashion industries.
Once the people in the portraits are acclaimed the portrait photographer will end up being acclaimed too.
Getting started can't be easy and I suspect it is not a smooth path to fame, relentlessly chatting up celebrities to come willingly before the camera, but a solid portfolio of past successes can definitely reinforce the "famous portrait photographer" profile.

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In today's world, good luck with getting to photograph any high-profile people. They usually have managers or agents and nothing gets done without them. But if you can photograph people with interesting faces, and develop a knack of getting the right expression and composition at the right time, that's a good start. It also helps to be able to either light or use natural light to its best. Study the portraits you like best, in books and online, practice on friends and family. Don't use mannequins or display heads. They never look good and won't give you any experience you'll need for working with real people. Diffused window light coming from one side is always an easy one.

I respectfully submit
"Light Years", by Douglas Kirkland.

warden said:

Arnold Newman is not recent, but I think careful examination of his portraiture will answer your question about what makes certain portraits or artists stand out. Enjoy.

https://arnoldnewman.com/portraits.html

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It is not just the subject in Newman's portraits, that stands out, but his bold composition and lighting.
As @warden has said, examine his work and you will see why he was mentioned. He may not be recent but good portraiture is like good music, it will stand the test of time.

One of my favourites of his, is this one.

I like Philippe Halsman. He took some wonderful straight portraits and also some very inventive ones.
Another good studio portrait photographer was Robert Mapplethorpe. And while you may not get good practice using mannequin heads, it seems you can learn a lot by taking pictures of flowers.
Diane Arbus used ambient light for her photos - I think possibly all the time. Andy Warhol mostly used a few Polaroid cameras with flash. You end up with totally different types of pictures and perhaps Big Shot type flash portraits are more mundane now than they were 50 years ago.
Fighting against the practised face that seemingly everyone now has for a camera - if you can get someone to drop that and not look like they're sneezing, I applaud you. People have so fully cemented their self-image at this point...

The National Portrait Gallery in London put out a book a few decades ago that is a giant well printed coffee table book FILLED to the gills with portraits, some recognizable by people we know of today and some by people who were well, known then but unknown now. I got my copy from a 2nds/remainders shop (do those stores even exist today ? ). there were always great photography books that typically cost an arm and leg there, for not too many clams. I can't remember the name of the photographer that did it, but one of my favorite portraits is one taken at a shipyard, with this person standing by these enormous links of a chain. Probably a salt print or albumen from glass negative or something, it was just perfect.

If you want to have your head clear when evaluating portraits, stay away from ones where it's either a celebrity in it or one behind camera. I say this as I assume the question is about pure visual impact / value that makes it stand out, not it costs now and will appreciate to, neither a determinant of what it actually looks like.

Also, Leibovitz was mentioned here, I'd put here at a bottom of the pack regardless. Andy Warhol ??? That is pretty much what I am alluding to at the start of this post.

I do agree that taking a look at Jane Brown might help. Certainly far above Leibovitz IMO.

Better yet, scan web and YT. Thousands of outstanding portraits are taken worldwide every day, many get posted on the net daily as well. The whole spectrum of worst to best gets filled more and more and gets ever more interesting.

Key is to view it without having any clue who is in it or who is behind it.

bilsmaron said:

I'm reading Susan Sontag book "On Photography" and looking at the works of the artists mentioned in the book like Diane Arbus and Andy Warhol, Arbus photos has something about them that I can't explain, but to my untrained(I'm a new hobbyist that does landscapes mostly) eye some of Andy Warhol portraits look ordinary.

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You're perceptive, as not many people get this. Warhol was a plain vanilla guy despite all the hype around him, and he was fond of saying that there was no underlying message in his work, it was all shallow and surface, just like him. The portrait models were often silk screen prints of movie stars (unreal, no substance), but he usually made Polaroids of them first and worked from them. Warhol was first and foremost always an artist, not a photographer.

Arbus was a photographer, and a great one. She liked to use people who had disabilities for her portraits. Their "otherness" comes through, even in a photograph. Perhaps, especially in a B&W photograph.

Supposedly it boils down to emotion.

Takes a lifetime to understand that technique and the gear is secondary.