TL;DR: Because Luke is a Jedi and Leia isn't.
However, the First Order is eager to kill both of the siblings - and in The Force Awakens, they actually come much closer to killing Leia than killing Luke.
Note: Italics in the following quotes are in the original; bold is mine.
What does the First Order have planned for Leia?
They intend to kill her, along with the rest of the Resistance. She's their enemy. But Luke poses a different, possibly bigger long-term threat. They use the Starkiller to destroy the seat of the New Republic government in the Hosnian System, then aim it at D'Qar, the headquarters of the Resistance - which is where Leia happens to be. If this attack had succeeded, Leia would have died.
Why does the First Order focus on Luke in particular?
Because he is a Jedi. Leia could have been a Jedi, but didn't become one. If Leia survives but Luke dies, Leia remains a lesser threat (at least in terms of Force potential), because she's just an untrained Force-sensitive. Luke is a trained Jedi, so if he survives, he can rebuild the Jedi Order. Leia can't, because she doesn't know much about the ways of the Jedi.
Could Leia even become a Jedi at this point?
We don't know, but she's well past the prime age for beginning Jedi training. In The Force Awakens, she's probably about 50 years old.
During the Old Republic, when the Jedi Order was going strong, kids began training when they were toddlers. Even Anakin, who was 9 years old when he met Qui-Gon and Obi-Wan, was initially deemed "too old" to begin training:
QUI-GON: I'm afraid not. Had he been born in the Republic, we would have identified him early, and he would have become Jedi, no doubt... he has the way. But it's too late for him now, he's too old.
...
OBI-WAN: The boy will not pass the Council's tests, Master, and you know it. He is far too old.
...
MACE WINDU: He is too old. There is already too much anger in him.
They trained Anakin despite these reservations, and what happened? He became evil, joined the Sith, helped kill Mace Windu, slaughtered all the younglings, helped Palpatine hunt down the other Jedi, and established himself as the second-worst person who ever lived.
In Episode V, when Yoda tells Obi-Wan that, should Luke fail, they can turn to Leia, Luke and Leia are in their early twenties. Yoda is hesitant to train Luke at first for just this reason:
Yoda: He is too old. Yes. Too old to begin the training.
Yoda trained him anyway, and what happened? He disregarded Yoda and Obi-Wan's pleas to resist the urge to save Han and Leia from Cloud City, lost a hand, left his training incomplete, ignored Obi-Wan's advice that he needed to kill Vader (although that part worked out for the best), almost fell to the Dark Side, survived only because Vader took pity on him, set up a Jedi Academy of sorts, saw his star pupil (and nephew) kill all the other students, abandoned the way of the Force, and went into hiding.
The moral of the story: When people begin training too late, it often leads to problems.
What has Luke taught Leia?
Not much. Mostly how to meditate and clear her mind:
The room is white and mostly empty. The walls are padded. The windows are many, and the sunlight streaming in is bold and bright.
The only things in this room are Leia and a potted plant.
The plant is a sapling of the sanctuary trees of Endor, though some call it a serpent’s puzzle, named so after the way the dark branches weave together in a kind of organic knotwork.
She grew it from a seed— a small knobby acorn given to her by the little Ewok known as Wicket. She grew the plant in a pot of Chandrilan soil, and to her shock and delight, it took.
It has become a focus of her meditations, as suggested by Luke. She decided, after storming out of the meeting room, that it was best to come here. Best for her to focus on something that wasn’t the state of the galaxy, or the nascent New Republic, or that nagging feeling in the deep of her middle that Mon has betrayed her in some small but significant way.
She sits with it in the middle of the room.
She clears her mind.
And then she tries to feel the tree.
She does this at least once a day.
Leia has never felt the tree.
Not for lack of trying! She sits here. She empties herself of breath, and then she tries to free herself of thought. Just like Luke taught her to. That part works fine most of the time. But he said it was possible to feel the lifeforce of things with the Force...
Luke continues to swear that, with time, she will come to feel the Force just as he does. He explained that it was how she felt his pain back during Cloud City— him hanging there, wearied and beaten and about to fall into the roiling clouds below. He said he’d teach her.
And he did teach her. Some things, at least.
Then? He left.
Just like Han left.
Luke...
- Star Wars: Aftermath: Life Debt
...
After a slow exhale, Leia says: “My brother taught me to center myself. To be mindful of what I’m feeling— a cup to be filled up, he says.”
- Star Wars: Aftermath: Life Debt
Why didn't Leia devote herself to becoming a Jedi?
We don't know all the reasons, but a few are mentioned.
Some of the reasons are revealed in two canonical novels - Aftermath: Life Debt, set in 4-5 ABY, about a year after Endor, an dBloodline, set in 30 ABY, about 26 years after the Battle of Endor.
“Did you never consider following in your brother’s path and becoming a Jedi?”
Leia found herself caught short. “Why do you ask?”
“They say on my world that the Force sometimes runs strong in certain families.”
So much of the lore of the Jedi had been lost— but on Gatalenta, the old religion had remained strong. History had become legend, but some of the legends were still told. Gatalenta had been one of Luke’s first destinations when he began his research into the Jedi Knights of old. Tai-Lin continued, “If that is true, then you might have the potential, just like your brother.”
And my son, she thought but did not say.
“If you have that ability, then I cannot imagine why you would not become a Jedi as well,” Tai-Lin finished. “Surely I’ve known few people who would make a finer make a finer Jedi Knight than you.”
Leia inclined her head in gratitude for the compliment, but she could not answer right away, because she could not tell the full truth1. The Force was too important a subject to be shared lightly, even with Tai-Lin, her ally and friend. Her safe, sensible, and, as far as it went, honest reply: “My duty has always been here, in the work of creating a new and better government.”
- Star Wars: Bloodline
...
She swore to [Luke] that she just doesn’t have it. It being that mystical, intangible power that her brother possesses and (this thought comes with a set of chills grappling up her spine) that her father— her birth father— possessed, too.
- Star Wars: Aftermath: Life Debt
...
Luke said those strong instincts of hers served as proof of the Force, evidence that it was working through her all the time. Maybe he was right. But Leia believed just as much in her experience and her common sense.
- Star Wars: Bloodline
...
...Luke Skywalker, too, but this detail was nearly irrelevant to Lady Carise. Skywalker had been so long away on his strange quest for the lore of the Jedi that he no longer had much influence outside his own acolytes. He was a figure of myth more than one of flesh and blood...
"Princess Leia spoke of her brother, the famous Luke Skywalker, who has been little seen in the public sphere for many years now."...
“Since the Rebellion, Skywalker has lived a private life. He has asked no more of the New Republic than any of its other citizens, nor have we just cause to ask more of him than the substantial service he has already given."
- Star Wars: Bloodline
...
“That tracks. Two heroes of the Battle of Endor, and I haven’t seen Solo around here in months. Skywalker for even longer.”
- Star Wars: Aftermath: Life Debt
Thus, Leia opted not to train as a Jedi because:
She wasn't convinced she could be a Jedi, and doubted her Force-potential
She wasn't sure she should be a Jedi, because Vader was her father2
She felt she was better suited for a political leadership role, and as such, became a New Republic senator
Luke disappeared shortly after the Battle of Endor to find Jedi ancient artifacts and sites. He returned briefly to take Ben Solo on as a student, then left again to train his younglings. After Ben turned to the Dark Side and killed the other younglings, Luke went into hiding on Ahch-To, and his location wasn't revealed until the end of The Force Awakens.
Notes:
1 i.e., that Vader was her father and this made her wary of becoming a Jedi for some reason
2 She never explicitly explains why she doesn't apply this reasoning to Luke, since Luke is also Vader's child, but it might have something to do with the fact that Luke, unlike Leia, witnessed Vader's redemption firsthand:
As always when Leia thought about this, she called upon what Luke had told her of their father’s last hours. He had renounced darkness, saved Luke, and become Anakin Skywalker again. Whenever Luke told the story, a beatific smile lit up his face; his memories of that event gave him a level of comfort and even joy that sustained him. Those were memories Leia couldn’t share...
Whenever she’d talked with Luke about their birth father, this was the part where he’d refuse to use the name Darth Vader. He was Anakin Skywalker when he fell in love with our mother, Luke would say, taking her hand gently in his. And he became Anakin Skywalker again in the last hour of his life. He came back from the dark side, Leia. They said it could never be done, but our father did it. He made that journey because of his love for us.
Leia believed Luke. She could feel that truth within him. But it was difficult for her to find solace in this the way Luke did. How could Vader torture her without mercy if he had that good inside? He’d still had the power to make the right choice, but had instead forced her to suffer...
Think of your conversation with Casterfo as practice, she told herself. One day she would have to reveal all this to her son. The truth of Vader’s identity had shattered her; she could not imagine what it might mean to Ben. At least Luke could tell Ben the most important part— that Vader had, in the end, been redeemed. Anakin Skywalker had returned; the dark side had been defeated by the light.
Leia knew this. She believed it. But she still did not understand it.
- Star Wars: Bloodline