well respected but not viewed as “hero material” by most of the court. When he accepts the Green Knight’s challenge and journeys to the Green Chapel, most do not expect him to return. Fortunately for Gawain, the Green Knight’s tests are moral rather than physical. Although Gawain lies and fails one of the Green Knight’s tests, due to his moral fortitude, he escapes with his life and returns to Camelot.

Despite their differences, Sir Gawain and Beowulf share many similarities. Beowulf and Gawain are heroes, even if the exact nature of their heroism varies. Both characters also embody the “good” in two opposing value systems. Beowulf is a “good” warrior and loyal fighter and, by extension, a “good” man in the context of a warring Anglo-Saxon culture. Sir Gawain embodies a more peaceful and Christian “good” that became prominent in the Chivalric Code of the early 1200s. Displaying non-violent virtues such as honesty, loyalty, and sexual purity, Gawain’s morality is what makes him heroic. Finally, both Gawain and Beowulf are honored for their work and receive acolytes from their kings. Beowulf’s funeral is described as the largest in history while Gawain’s green sash is adopted by the Knights of the Round Table to remind themselves of the importance of honesty.

I hope this helps!

The villains in Beowulf are Grendel, Grendel's dam (mother) and the dragon that finally kills Beowulf—extremely evil characters who want to wreak havoc and destroy humans. The Christian influences in this surviving version of Beowulf blames the evil inherent in Grendel on the sin of Cain in the Bible's Old Testament.

Grendel was this grim beast called, who haunted the moors and secluded fens; this accursed one had long dwelled with monsters since the Creator had decreed his exile. On the kin of Cain did the sovereign God avenge the slaughter of Abel; Cain gained nothing from this feud and was driven far from the sight of men...From him awoke all those dire breeds: ogres, elves, and phantoms that warred with God a lengthy while...

Grendel he has been punished—separated from the company of man and God through the sins of Cain. This fuels his hatred, and a desire to destroy goodness from the world of which he can have no part. 

When Beowulf fatally wounds Grendel, Grendel's dam returns for revenge. She is motivated by hatred—hoping to destroy the humans who have killed her son.

The hag came to Heorot, where the...Danes slept in the hall. The princes' old woes came back suddenly when Grendel's mother burst into their midst...That hag was in haste, wanting to flee with her life when the liegemen spotted her...she seized a single clansman [and] fled to the moors. 

Beowulf and Hrothgar travel to retrieve Hrothgar's favorite—the liegeman the monster took. Beowulf valiantly fights and kills the dam.

Many years later—a thief disturbs the hidden treasure of a dragon that then attacks Beowulf's kingdom, killing his people. 

When the dragon awoke...He immediately sniffed the scent along the stone. The dark-hearted one found the footprints of that foe who had walked undetected by the creature's head...The guardian of gold went tracking over the ground, eager to find the man who had brought mischief upon his slumber. Savage and burning, he circled 'round the barrow...he desired war and was eager for combat. 

Beowulf valiantly saves his people, but dies from his wounds.

The villains in Beowulf are murderous. However, the Green Knight issues a challenge and Gawain accepts it in Sir Gawain and the Green Knight—but the giant has no wish to harm. He appears in Arthur's court at Christmas and issues a challenge: anyone can try to cut off his head. Young Gawain tries to do so; in failing, he is called to face the green creature in a year to give the Green Knight his opportunity to take Gawain's head, as a part of their agreement.

A year later, traveling to fulfill his promise, Gawain meets Bertilak (really the giant) and Lady Bertilak. He is tested secretly so Bertilak can see if Gawain is an honorable knight in Arthur's court. The giant spares Gawain, for he was honorable—only trying to trick the Green Knight in order to save his own life. But Gawain is appalled at his actions:

Oh knight: I humbly confess

My faults: bless me

With the chance to atone.

"I'll try to sin less."

Bertilak holds no grudge; he has convinced Gawain to work harder to be a worthy part of King Arthur's court. 

Where Beowulf faces villains intent upon killing humans, Gawain faces a creature intent only in pushing the young knight to more fully know himself, so as to better serve his King.

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

http://www.enotes.com/sir-gawain-and-the-green-knight/historical-background