In "Gawain and the Green Knight," Gawain is a young, but impetuous and untried knight of King Arthur's court who accepts the challenge of the Green Knight to do battle. When Gawain loses, he promises that he will meet the Green Knight in one year so that the Green Knight has the chance to exact his revenge from the young man.
When Gawain arrives at the agreed upon location, temptation is placed before him three times in order to test his honor as a true and loyal knight. This is the central focus of the story.
The difference between this popular version and what seems to be the original source of the story is that "Gawain and the Green Knight" includes medieval values and moral codes.
The earlier stories are considered "beheading game" stories that did not focus on the elements of chivalry (knightly virtue, honor and courtly love) as the "Gawain" story did. The early stories focused on the battle—the beheading.
So the primary element that differentiates the old beheading game stories, and "Gawain and the Green Knight" story is the presence of chivalry in the second tale.