I didn't understand the difference, so I finally did a search on the two titles, "King in the North" and "King of Winter."
Based on a few of uses of the phrase "King in the North" I had noticed before doing the search, I expected to find that "King in the North" was an incorrect title, and might explain why Robb's reign didn't last long - he never set foot in the north after being crowned, he had been born at Riverrun and just didn't qualify as a true King of the area north of the neck, in my thinking. If one of his brothers later becomes King, I expected we would see them adopt the title "King of Winter" and be more successful than Robb had been.
Instead, looking first at only AGoT search results, I found an apparent distinction between living and dead kings. Characters discussing the historic Stark kings and their battlefield activities or other history would refer to the rulers as Kings in the North. Characters in the Winterfell crypt, looking at the swords across the laps of the stone likenesses of each man, would use the term Kings of Winter.
Somehow I know I have to go down there, but I don't want to. I'm afraid of what might be waiting for me. The old Kings of Winter are down there, sitting on their thrones with stone wolves at their feet and iron swords across their laps, but it's not them I'm afraid of. I scream that I'm not a Stark, that this isn't my place, but it's no good, I have to go anyway ...
(AGoT, Jon IV)
The first Lords of Winterfell had been men hard as the land they ruled. In the centuries before the Dragonlords came over the sea, they had sworn allegiance to no man, styling themselves the Kings in the North.
(AGoT, Eddard I)
But then there's this, near the end of AGoT:
Maege Mormont stood. "The King of Winter!" she declared, and laid her spiked mace beside the swords. And the river lords were rising too, Blackwood and Bracken and Mallister, houses who had never been ruled from Winterfell, yet Catelyn watched them rise and draw their blades, bending their knees and shouting the old words that had not been heard in the realm for more than three hundred years, since Aegon the Dragon had come to make the Seven Kingdoms one … yet now were heard again, ringing from the timbers of her father's hall:
"The King in the North!"
"The King in the North!"
"THE KING IN THE NORTH!"
(AGoT, Catelyn XI)
Someone had pointed out to me in another thread that Mormont's voice was different here. Why is Maege Mormont using the term associated with a dead King, while the other northern bannermen are using the term for a living Stark king? Do the Mormonts know something about the Starks, and/or about the north, that would cause her to choose that title instead of the title everyone else is shouting?
And then there's this insight from Osha, who seems to know a lot about the north and old gods and lore and legend (or history, depending on your perspective):
He wished they were here now; the vault might not have seemed so dark and scary. Summer stalked out in the echoing gloom, then stopped, lifted his head, and sniffed the chill dead air. He bared his teeth and crept backward, eyes glowing golden in the light of the maester's torch. Even Osha, hard as old iron, seemed uncomfortable. "Grim folk, by the look of them," she said as she eyed the long row of granite Starks on their stone thrones.
"They were the Kings of Winter," Bran whispered. Somehow it felt wrong to talk too loudly in this place.
Osha smiled. "Winter's got no king. If you'd seen it, you'd know that, summer boy."
(AGoT, Bran VII)
So I pressed on and looked at the search results for each royal title in ACoK, to see if the distinction was any clearer. We don't have as many scenes or flashbacks to the Winterfell crypt, so the use of the phrase "Kings of Winter" are fewer in number. But the ones that occur don't seem to follow the pattern more-or-less established in the first book. First Catelyn in a POV and then a line of dialogue from Jaime to Catelyn:
The ancient crown of the Kings of Winter had been lost three centuries ago, yielded up to Aegon the Conqueror when Torrhen Stark knelt in submission. What Aegon had done with it no man could say. Lord Hoster's smith had done his work well, and Robb's crown looked much as the other was said to have looked in the tales told of the Stark kings of old; an open circlet of hammered bronze incised with the runes of the First Men, surmounted by nine black iron spikes wrought in the shape of longswords. Of gold and silver and gemstones, it had none; bronze and iron were the metals of winter, dark and strong to fight against the cold.
(ACoK, Catelyn I)
"My son may be young, but if you take him for a fool, you are sadly mistaken . . . and it seems to me that you were not so quick to make challenges when you had an army at your back."
"Did the old Kings of Winter hide behind their mothers' skirts as well?"
"I grow weary of this, ser. There are things I must know."
(ACoK, Catelyn VII)
Catelyn is recalling the historical event of a living Stark king surrendering the crown to the living Aegon the Conqueror, so why is the crown described as being from the Kings of Winter? Jaime's choice of words might just be a mistake because he is not from the north and might not be clear on the distinction. But he does know enough about the north to later explain to Brienne that the Boltons used to flay the Starks and make cloaks of their skin. These examples seem particularly important in sorting out why GRRM would use two different terms to describe the traditional Stark kings.
The Bran VII chapter of ACoK has references to both types of kings, and they don't seem to follow the living / dead distinction. First, Bran describes crypt Starks using the term I tought denoted living Starks:
Their footsteps echoed through the cavernous crypts. The shadows behind them swallowed his father as the shadows ahead retreated to unveil other statues; no mere lords, these, but the old Kings in the North. ... He had never feared the crypts; they were part of his home and who he was, and he had always known that one day he would lie here too.
But now he was not so certain. If I go up, will I ever come back down? Where will I go when I die?
Bran did say a page earlier that, "It looked for an instant as if the dead were rising" with the movement of shadows cast by a newly-lit torch. So maybe the message is that these dead Starks have now come to life. I have noticed a few references to a sword as a shadow. If swords and shadows are supposed to be compared, it would make sense that movement of shadows in the Winterfell crypt might free the spirits of dead Starks: their tombs are built with swords across the laps of each lord's statue, with the iron in the sword ensuring that the spirit of the dead person can't wander out of his crypt. Moving shadows could symbolize moving swords, freeing those spirits. Bran and his party remove four literal swords from the tombs just before the excerpt I cited, above. The swords belong to Lord Eddard (taken by Osha), Lord Richard (taken by Meera) and uncle Brandon (taken by Bran). Hodor takes an old rusty blade from an unidentified tomb.
The cited passage also includes Bran referring to himself, to feeling at home in the crypt and to speculation about his own interment. So maybe GRRM uses the "living" king title because Bran is about to put himself in the context of the long line of Stark kings. At the end of the chapter, however, Bran refers to himself as living but switches to the use of the "Kings of Winter" title to refer to his predecessors:
... it was hard to tell that the castle had been sacked and burned at all. The stone is strong, Bran told himself, the roots of the trees go deep, and under the ground the Kings of Winter sit their thrones. So long as those remained, Winterfell remained. It was not dead, just broken. Like me, he thought. I'm not dead either.
Instead of the difference between living and dead kings, could the difference be between kings who fight men and kings who "fight against the cold," as Catelyn says in her description of the crown? Do all dead Stark kings fight against Winter? Is that what the afterlife is like for Starks?
I may try to follow up and search the "Kings of Winter" and "Kings in the North" phrase for the remaining books, to see whether the pattern becomes clearer. Or feel free to share examples or evidence if you think you've discovered why there are two different titles.