Interesting question! The biggest difference I immediately notice is the time period in which each story was written. A manuscript of Beowulf exists, but it was centuries old when it was finally recorded— it is anonymous, and it was passed down before that through bards, or storytellers. Sir Gawain and the Green Knight is also anonymous, but is a "younger" piece, believed to have been written at the end of the medieval period, as feudalism was on its way out with the advent long bows, gunpowder, cannon, muskets, etc.
Beowulf includes references to God (Christianity) and homage is paid to Him. Also, Grendel is said to have descended from Cain, Adam and Eve's son who was cursed by God for murdering his brother, Abel. Also:
[Beowulf includes many] juxtapositions of pagan and Christian references. The names of the heathen gods have been omitted from the poem itself, but there are still many references to Fate (Wyrd) or destiny, and the author seems to simultaneously extol both the pursuit of worldly fame and the reliance upon Providence/God.
It is possible that these "juxtapositions of pagan and Christian references" are the result of transposing a tale originally told before Christianity reached the first person to tell this story, to a tale that reflected a world that now included a belief in God.
Christianity is also referred to in Sir Gawain and the Green Knight—first, it takes place at Christmas. Secondly, knights were supposed to be servants of the Church, something the Catholic Church had instituted in order to control these warriors, especially after their vicious and brutal behavior during the Crusades, the intent of which was to save Jerusalem from the "infidels." (See chivalry.)
However, there are also pagan references in the story of Gawain.
[Gawain] later retains some of the roughness of his Pagan heritage. In the Arthurian romances of Cretien de Troyes, Gawain is associated with magic and unrestrained sexuality...In Mallory’s Mort D’Arthur and other romances, Gawain is given a feature that may go back to a Pagan deity associated with the sun. His strength increases, like that of the sun, in the morning, peaks at noon and then declines.
The Green Knight also contains references that are more pagan in nature:
Scholars have long debated whether [the Green Knight] owes more to Pagan mythology...[Like a tree,] The Green Knight has [the regenerative nature of a tree.] On being decapitated, he simply picks up his head, which continues to speak in his hand. The next year, the head is back on his torso where it belongs.
The similarities of the codes each character follows can be found in the heroes' brave and honorable nature. Each has a code that guides his actions: each is ready to sacrifice his life for the sake of his honor. Each man is aware of things in the world far greater than himself, and shows humility.
Although the stories take place in different locations, and spring from different eras, the ideal of the hero seems much the same.—the concept is timeless. And though their stories are decidedly different, their actions and values (their codes) seem particularly similar.
Additional source: http://www.medieval-life.net/chivalry.htm