1. In a Song of Ice and Fire, there are more Wildings dead outside the wall than in a modern day cemetery. Ygritte was Jon Snow’s love, but he chose his duty over her. She was this lively redhead who made Jon Snow break his oath.
Had she not been killed in season 4 by Olly, she and Jon Snow would have patched up their misunderstanding and when Sansa comes over to Winterfell they would have got to know each other. But with Ygritte in the Wall, Jon would not have been there for a long time as he had broken the oath he took as a Night’s watch.
But had he stayed at the wall, he would have definitely been killed. That occurred as a result of bringing the wildings to stay on the wall. After he had been resurrected by Melisandre, he would have fought Ramsay but would not have taken Ygritte into the war.
She would have been by his side after he had conquered Winterfell. He would have shown her around the castle and she would have finally seen what a castle looks like. She would have seen the rooms where Jon grew up and would have been along with Sansa. They would have probably bonded together. There would have been one pair in Westeros who truly loved each other.
2. Had Ygritte lived she would not have been able to live with Jon Snow South of the Wall. She had always wanted him to stay with her North of the Wall. Maybe after Arya comes back to Winterfell, she would have liked Ygritte a lot as they both have a lot in common. They both defy the male and female discrimination that happens in Westeros and Arya is a wilding at heart, so it would have been a perfect meet-up. The incident where Jon Snow is killed never happens as Ygritte would see through things which Jon Snow could have never acknowledged. Jon is always known to make poor choices and she would have been for him to correct him. Had Jon not been stabbed, things would have been different a lot. But had Ygritte been alive, Jon would have never become Lord Commander of the Night’s watch.
This means that the peace between the Night’s watch and the wildings would have been a lot harder to come by as Alliser Throne would have been the Lord Commander. But this would not have been impossible as majority of the Night’s watch support Jon Snow.
3. After Ygritte dies Jon was acting withdrawn from everyone even after becoming Lord Commander. At every phase of his life, there was something that reminded him of her. Every time someone said “You know nothing Jon Snow”, it was always Ygritte. For Jon Snow, things would have been a much happier one had she lived. She would have been with him to keep him happy in his dull and gloomy life. But Jon marrying a wilding who knew nothing about the world beyond the Wall was a question. Would he marry her? Or always have her by his side? As a wilding she wanted Jon to be with her, the North of the Wall. But with her, we saw a completely different Jon, one who was laughing (finally!).
Had Ygritte been alive, things wouldn’t have changed much at all. Jon still would have been stabbed to death. That might have happened even earlier with Ygritte being alive. But Jon Snow would have been happier with her around. But had she survived Jon might have gone over the Wall to live with Ygritte in a wilding village. He might have been happy, but he wouldn’t have been the Lord Commander or would have become the King in the North.
Maybe as Ygritte always said they should have stayed at the Cave.