Throughout history, different cultures have held different virtues in the highest regard. The Anglo-Saxons, like the anonymous scop of the poem Beowulf, valued strength and courage over any other quality. On the other hand, the earliest English people, like the poet of Sir Gawain and the Green Knight, respected honor and the basic codes of chivalry over all else. Both Beowulf and Sir Gawain, therefore, display the most favored qualities of their people: Beowulf by defeating monsters, performing great feats of courage, and telling of his deeds; Gawain by being loyal, brave, and honest. Both sets of qualities are displayed throughout each tale.
In Beowulf, written in about the fourth century AD, the hero's first major test is to fight a huge monster. In doing so, he proves his physical abilities, and thus establishes his hero/ leader status. To be the best, you have to be the strongest, and he says "Fate saves/ the living when they drive away death by themselves!" (572-73), showing how much Beowulf relies on his physical strength.
On the other hand, Gawain's first test is not physical, but psychological. He is forced to prove his loyalty to Arthur, by offering his life to the Green Knight. He does not claim to be the best, or the strongest, and in fact claims: "I am the weakest, I am aware" (part 16). Gawain is displaying his modesty, and the medieval belief that a person should be humble about himself. This displays the immediate difference between the two cultures. For the Anglo-Saxons, if you wanted people to know you're a hero, you tell them you're a hero. For Gawain, and the Knights of the Round Table, they must act humbler, in order to appeal to their Christian beliefs.
However, both societies do require their heroes...