What are the similarities and differences between the societies in Beowulf and Sir Gawain and the Green Knight?

Comparing the differences between the societies depicted in these two works brings us to a few small problems with a sociological analysis of them. Both Beowulf and the tale of the Green Knight occur in a fantastic realm where story carries more commentary on reality than reality itself. Both provide lessons about heroic acts and choices, offering suggestions to their audiences about how to be a good man. Inferences can be made about the societies that created these myths, even if it may be unproductive to study the limited, fantastic social worlds they contain.
Both the society of Beowulf and the society of the medieval romances believed that bravery and honesty were crucially important to survival. The character of Beowulf, for instance, shows up Unferth, a rival warrior, by pointing out the dishonesty of his words. Beowulf's boasts are grounded in truth, while Unferth skews the truth to make Beowulf look bad. Ultimately, Beowulf's word proves trustworthy, and so Unferth is the one who loses face. Gawain, likewise, makes a point of showing up for his appointment with the Green Knight. An Arthurian knight should keep his word, and he does so. Both Beowulf and Gawain face their enemies fearlessly because they have said they would do so, displaying both the truth of their promises and the courage of their characters.
One glaring difference between the two stories is that in Beowulf, warriors love to boast of their successes in battle, while in the Gawain story, humility is considered wise and honorable. This may relate to the influence of Christian theology. Christianity had only recently been introduced at the time Beowulf was written down, and so we see a mix of Christian and pagan thinking in the epic. In the Middle Ages, however, Christian thinking had long since come to dominate English culture, and so humility (as expressed in Jesus's "Sermon on the Mount") has a more pervasive influence on the literature.
The heroes' reasons for fighting are also quite different. Beowulf goes to Denmark to face Hrothgar's monster because it threatens people's lives, and he faces the dragon in his homeland later in life to save his people from the monster's threats. Gawain's fight, on the other hand, does not stem from the need to protect anyone. The story begins with Arthur asking for an exciting tale from one of his knights' experiences, and the challenge offered by the Green Knight is one Gawain willingly accepts just for the sake of proving himself. No one is in danger from the Green Knight, but Gawain agrees to fight him anyway. When we consider the difference in years—700–800 CE for Beowulf and 1200–1400 CE for Sir Gawain and the Green Knight—it becomes clear that the political and social landscape drastically changed within those several centuries. Ask yourself what type of political context creates a story in which all of the threats are coming from the outside and, if not met, will destroy everyone. Conversely, what type of political context creates a story in which a warrior chooses battles that offer no needed protection but rather a valuable moral lesson?