I was just watching Goblet of Fire and remembered something Voldemort said in The Deathly Hallows-1. He said that "My wand and Potter's share the same core. We can wound but not fatally harm one another". But in the graveyard, he was able to torture Harry and tried to kill him using AK, and Harry's wand did nothing. Why so? Also, since it was Voldemort dueling Harry, Harry's wand should have done that golden flames thing to prevent Voldemort from killing or torturing Harry, but it didn't. Is there an explanation for this?
Since you mention a quote from the film version of Deathly Hallows, I shall mostly focus on the movies.
In the film Harry Potter and the Deathly Hallows: Part 1, Voldemort does indeed say that he can harm Harry with his wand, but not kill him:
VOLDEMORT: But I face an unfortunate complication. My wand and Potter's share the same core. They are, some ways, twins. We can wound but not fatally harm one another. If I am to kill him ... I must do it with anothers wand.
In the original novel, the scene is a little different:
“As I was saying,” continued Voldemort, looking again at the tense faces of his followers, “I understand better now. I shall need, for instance, to borrow a wand from one of you before I go to kill Potter.”
The faces around his displayed nothing but shock; he might have announced that he wanted to borrow one of their arms.
“No volunteers?” said Voldemort. “Let’s see . . . Lucius, I see no reason for you to have a wand anymore.”
Lucius Malfoy looked up. His skin appeared yellowish and waxy in the firelight, and his eyes were sunken and shadowed. When he spoke, his voice was hoarse.
“Your wand, Lucius. I require your wand.”
—Harry Potter and the Deathly Hallows
This is partly because of Priori Incantatem. If two spells from wands that share a core collide, a battle commences between the two wielders, with the stronger of the two forcing the weaker to recapitulate in ghostly form the spells it has cast:
“So what happens when a wand meets its brother?” said Sirius.
“They will not work properly against each other,” said Dumbledore.
“If, however, the owners of the wands force the wands to do battle . . . a very rare effect will take place. One of the wands will force the other to regurgitate spells it has performed — in reverse. The most recent first . . . and then those which preceded it. . . .”
—Harry Potter and the Goblet of Fire
This is why Voldemort needed to use Lucius's wand in the book continuity: if he and Harry had tried to cast spells at each other, they might have connected, rendering Voldemort's spell ineffective and preventing him from effectively casting others.
The movie continuity seems to imply something different: that Priori Incantatem will only occur when Voldemort tries to kill Harry. Regardless, it seems likely that Voldemort is talking about the effect that occurred in Goblet of Fire, as opposed to an entirely new perk of having twin cores.
In any case, when he tried to kill Harry in the graveyard, he was unable to kill him. The first Avada Kedavra he cast missed, and the only one that could have hit was blocked by Priori Incantatem
Why didn't it shoot golden fire?
This was an entirely separate effect.
Voldemort did not completely understand the nature of the connection between him and Harry. Even when he used Lucius's wand, the connection between him and Harry still protected the latter. From the screenplay:
Abruptly, Harry’s arm rises as if on a string, drawn up by the wand trembling in his hand. GOLD FIRE spits forth and -- Crack! -- SPARKS skitter up the chassis of the motorbike. Harry wheels and -- for a split second -- is face to face with Voldemort, whose eyes drift, staring -- with something like fear -- at Harry’s wand. Then -- whoosh! -- Voldemort peels back.
The book describes the reason for this in more detail.
“I believe that your wand imbibed some of the power and qualities of Voldemort’s wand that night, which is to say that it contained a little of Voldemort himself. So your wand recognized him when he pursued you, recognized a man who was both kin and mortal enemy, and it regurgitated some of his own magic against him, magic much more powerful than anything Lucius’s wand had ever performed. Your wand now contained the power of your enormous courage and of Voldemort’s own deadly skill: What chance did that poor stick of Lucius Malfoy’s stand?”
“But if my wand was so powerful, how come Hermione was able to break it?” asked Harry.
“My dear boy, its remarkable effects were directed only at Voldemort, who had tampered so ill-advisedly with the deepest laws of magic. Only toward him was that wand abnormally powerful. Otherwise it was a wand like any other . . . though a good one, I am sure,” Dumbledore finished kindly.
—Harry Potter and the Deathly Hallows
When Dumbledore say "that night," he refers to when Harry and Voldemort's wands connected. At the time of the event mentioned in the question, Harry and Voldemort's wands had not yet experienced Priori Incantatem. Not having "imbibed" Voldemort's power, Harry's wand could not use it against him.
Wands aren't sentient; it can't do anything to actively "protect" Harry.
However when used against each other, this is when the "protection" occurs. Regardless of the spell used, the Priori Incantatum spell is created by creating a (seemingly kinetic) bond of energy between the two wands, causing the spells used previously to be negated.
When a duel occurred between wizards with wand cores harvested from the same creature, simultaneous spell-casting by both parties would trigger a rare, powerful effect called Priori Incantatem. Both wands became linked through a single, golden thread of energy. Hundreds of smaller golden beams shot off from this central thread to form a cage around the duellers.
What's required for this is for both users to cast a spell at each other (deflecting a spell doesn't count); and for them to cast the spells at the same time.
As for the Cruciatus Curse, it's a torturing spell - not something that would really be considered "mortally wounding". I.e. The Longbottoms were driven mad by the pain, but they are still otherwise alive and well. Additionally, I really don't believe that Voldemort wanted to really really hurt Harry using the Cruciatus Curse - he just wanted to see him squirm.
1Your first statement that wands aren't sentient is not exactly true. The wand chooses the wizard. Also, a wand will not perform well for you unless you win its allegiance. Apart from this, Harry's wand produced a spell all on its own to destroy Lucius' wand which was being wielded by Voldemort in DH. Wands are sentient to a some extent. Jun 15, 2016 at 5:14