Vader didn't sense Luke as someone special until he made the connection that Luke was his son. Before that he was just another person with a connection to the Force.
In contrast, Vader sensed his connection to Luke Skywalker almost immediately. How did Vader detect his relationship with one of his offspring but not the other? The short answer is that creator George Lucas had no idea Leia would end up becoming Darth Vader's daughter.
In this comic issue, Darth Vader is informed by Boba Fett that the Rebel pilot who destroyed the battle station (and briefly battled Vader on Cymoon 1 in issue 2) is named Luke Skywalker, who Vader immediately recognizes as his son.
So, how did Darth Vader know that Luke was his son? Darth Vader found out that Luke was his son through Boba Fett, who revealed to him that the name of the Rebel that destroyed the Death Star was Luke Skywalker.
Vader had failed. He could never convince Luke to join him in overthrowing the Emperor and he likely doubted the Emperor could ever turn Luke to the darkside. So if he captured Luke then he would only ensure his son's death which would most likely come only after extensive torture perhaps over several years.
He wasn't supposed to kill Luke; he wasn't even supposed to beat Luke. Vader's only goal in the fight was to make Luke lose control of his emotions and bring him closer to the dark side -- and that's exactly what happened. Vader had to obey his master, so he goaded Luke until he touched the dark side.
As mentioned, it would seem that each generation of the Skywalker family becomes stronger than the last. After Anakin Skywalker failed to pursue his destiny, Luke Skywalker became the new "Chosen One," and thus the Force became even stronger in him than in his father.
A Skywalker by blood, Leia Organa is known to be Force sensitive. Is she showing those abilities in the new Star Wars miniseries? "I am inside your mind." These are the first words that the phony Jedi Haja Estree speaks in the second episode of Obi-Wan Kenobi, pretending to read the thoughts of a customer he's conning.
Darth Vader didn't learn he had a daughter until Return of the Jedi, when he stood before Emperor Palpatine. The Emperor's power probed Luke's mind, attempting to goad him to fall to the dark side, and he discovered Leia's existence. "Sister," the Emperor taunted.
The space station was destroyed, but Vader survived, and he flew into the far reaches of space. After the destruction of the Death Star, Darth Vader became obsessed with finding Luke Skywalker, who he discovered was his son.
Alderaan's princess knew she had been adopted by Bail and Breha Organa after the Clone Wars, but she had no idea who her parents were. That was until Luke Skywalker told her the truth - that he was the son of Darth Vader, and that she was his sister.
His memory was wiped at the end of Revenge of the Sith. A blabbermouth like him can't be trusted with sensitive information.
However, she chose not to tell Anakin, presumably because he was about to go into a battle. Anakin was prone to letting his emotions affect him, and knowing that Padme was pregnant would've thrown him completely off the task. And Padme probably didn't think that her husband would be fighting battles for months to come.
What would happen if Darth Vader came across R2-D2 and C3PO? He came across Threepio on Bespin. He either didn't realize it was the same protocol droid or figured he'd been memory-wiped so FUNCTIONALLY wasn't the same droid and didn't care.
Although we never see Leia fully reconcile with the fact that Darth Vader is her father, it's worth noting she did actually forgive and accept the former Sith Lord in the non-canon Legends novel "Tatooine Ghost," by Troy Denning.
She figured out her mother's identity after Return of the Jedi, when she learned she was the daughter of Anakin Skywalker; it didn't take Leia long to connect the dots between Anakin and the senator of Naboo he had spent so much time with during the Clone Wars, who died under mysterious circumstances on the day Leia ...
This is not because the Third Sister knows that Leia is Anakin Skywalker's child; it is because the Third Sister found a link between Obi-Wan and Bail Organa and she believed Obi-Wan would do anything to save Bail's daughter.
Theory 1: Leia Doesn't Remember Ben Is Obi-Wan
Throughout her encounters with Kenobi in the series, he refers to himself as Ben. The most straightforward explanation would be that Leia knows the person who rescued her is Ben, and never connected the dots that Ben is actually Obi-Wan.
Legend had it that Yoda—a Jedi who became Grand Master—was trained by N'Kata Del Gormo. A Force-sensitive Hysalrian, N'Kata Del Gormo was trained in the ways of the Force and achieved the rank of Master within the Jedi Order.
Though there was some debate among the fans that Anakin Skywalker's son, Luke Skywalker, was actually the Chosen One since he caused his father to destroy Darth Sidious, the debate was settled when George Lucas himself confirmed in an interview that Anakin, even after becoming Darth Vader, was still officially the ...
And with how little training Luke had received, compared to Vader's years of experience, the fight is over surprisingly quick. However, there are a number of reasons the fight went the way it did. Perhaps the most important of all is the conflict within Vader, as he had just had a chat with his son on Endor.