"Tropic of Eden" by Lee Killough. ISFDB says it was first published in The Magazine of Fantasy and Science Fiction (August 1977). I first read it in The Best Science Fiction of the Year #7, edited by Terry Carr and released in 1978.
Your memory is pretty accurate. The beautiful woman in question is named Eden Lyle. At one point, the narrator says:
She was not merely an actress, not merely one of the most beautiful
women alive; she was a legend. Eve and Lillith, Penelope and Circe.
She had been the guest of every world leader in the past decade and
slept, so gossip claimed, with half of them.
The first-person narrator is Drummond Caspar, commonly called "Cas." He is a sort of sculptor; specializing in shaping "tropic sculptures" which are made from mutated plant material and visibly react to people and sounds in the nearby environment. In other words, a tropic sculpture is a living, moving variety of sculpture, instead of being stuck in just one shape. So the title "Tropic of Eden" seems to refer to a piece of Cas's work which he creates for Eden Lyle in the course of the story. The creation process involves her presence so that the living sculpture will be partially shaped by her own personality.
The younger girl who lives with Eden Lyle is called Hebe. As you say, her hair is cut much shorter, and she doesn't wear makeup or show the same poise that Eden has, but underneath those details they are extremely similar in appearance. At first, there's a rumor circulating that Hebe is some sort of cousin whom Eden has taken in. Cas later learns differently:
I felt I could not just leave, though, and so I said, “It’s uncanny
how much you look like your cousin in this light. If you wore some
make-up and let your hair grow, the two of you would look like twins.”
Hebe’s eyes lifted to mine, dark, unreadable pools. “We are.”
I did not immediately understand. “Are what?”
“Twins. Not cousins.”
I laughed. “There’s just a few too many months’ difference in your
ages for you to be twins,” I pointed out.
“I’m a clone,” Hebe said.
I realized several moments later that my jaw was hanging and snapped
it back into place. I tried to talk. I did not succeed very well. “A —
I thought — cousins, I was told — why would Eden — ”
“I asked once,” Hebe said. She sighed. “She wouldn’t tell me why she
had me made.”
At one point, Hebe moves near the tropic sculpture which Cas has made for Eden, and it reacts as if it's reaching toward her, wanting to grasp her. Cas has never seen that exact reaction before, and it gives him food for thought. He does some investigating, and finally figures out the answer to Hebe's question about why Eden wanted a clone of herself.
He has traced the history of Dr. Hugo Ascher, a man who has recently insisted that Hebe needs to make a trip to a certain spa in Switzerland. Cas learns that Dr. Ascher is a surgeon who was sued for malpractice after a brain transplant failed. The implication is that Eden intends to go to Switzerland and have her brain inserted into Hebe's skull so that Eden can enjoy being young all over again. (And, presumably, could repeat the process again a few decades later?) Cas is angrily telling Eden what he's deduced when they suddenly hear a scream from Hebe, whom they hadn't realized was eavesdropping on their conversation.
Hebe runs away. By the time Cas catches up with her, she has already pressed her cheek against the hot metal of a spotlight.
She let it go, smiling. The entire side of her face was purple and
shriveled. “Too late,” she said triumphantly. “No surgery can make me
perfect again.” She swayed. “No use to her now.”