1. The Wall was built hundreds of years ago to prevent the White walkers from entering into the land of the humans. It serves some sort of magical barrier which does not let the white walkers inside it. It is believed that the wall was built with the help of giants and magic from the Children of the Forest is imbibed in it so as the creatures from the other realm cannot pass through to the lands. Benjen Stark who was killed by a white walker and during his transformation, was rescued by a Child of the forest and came in time to help Bran and Meera from the White walkers.
“The Wall’s not just ice and stone,” he told Bran and Meera. “Ancient spells were carved into its foundation. Strong magic. To protect men from what lies beyond. And while it stands, the dead cannot pass. I cannot pass.”
According to the books, a Song of Ice and Fire speaks about something called the Horn of Winter and a blast from that Horn can destroy the Wall. Now suppose there exists no Wall; there would not have been any distinction between the people present in North and South of the wall. There would not have been any Men of the Night’s Watch.
2. If there was no Wall, there would not have been anyone guarding the fort. Hence, Jon Snow would not have gone to the Wall. He would have stayed back at Winterfell. Ned would not have taken him to King’s Landing. When Robb leaves for the war against the Lannisters, Jon would have stayed on Winterfell along with Bran and Rickon. Hence, Theon would not be able to capture Winterfell with Jon in it. So, Ramsay also would not have come to Winterfell. Bran would not have gone in search of the three-eyed raven and would have stayed at the castle. This would ensure Sansa returning to Winterfell to her brothers after escaping the Lannisters. There the sister and brothers would be reunited. There would be no battle of the Bastards. Jon would have never met Sam or any other men from the Night’s watch. Aemon Targaryen would have been a Maester at King’s Landing in the rule of Robert Baratheon. Or he may have been killed in the battle of Robert’s rebellion, just like all the other Targaryens. Daenerys would have escaped with Viserys and gone across the Narrow sea, away from Westeros. The wall would have only affected much of the happenings in the North as it would be less and less in the South.
3. The wall served as a barrier to the land of the Humans from the creatures that lie beyond the Wall. If there is no wall, the white walkers would have crossed the lands and would make everyone their kin. There would not be an Iron Throne as someone would have taken away all the Valariyan steel swords from the throne to fight the White walkers. They would succeed in their war as the white walkers cannot withstand a blow from the Valariyan sword. This would have brought a new reign, a completely new houses in Westeros. There would not have been a Lannister house or Baratheon or Stark or Tully or Tyrell. But a new system, a newer form of reign would have taken place. This would have brought a whole set of new storyline for GRRM.
The Wall was originally built to keep the White walkers away from Human folk. Without it the land would fully turn out to be a dead man’s land. Until some hero saves them all.