How can one compare Beowulf to Sir Gawain and the Green Knight?
Although we study both Beowulf and Sir Gawain and the Green Knight as examples of "medieval English literature", they represent quite different time periods and cultural traditions.
Beowulf is written in Old English and is part of the Norse Saga tradition. Most of the action of the poem takes place in Sweden and Denmark, and the heroes are Scandinavian. The poem shows elements of oral composition, and is written in a strong-stress alliterative meter using a four-stress line. Its actual date of composition was sometime between the eighth and eleventh centuries. Thematically, it is an heroic epic.
Sir Gawain and the Green Knight was composed in the fourteenth century in Middle English (more than three centuries later than Beowulf), and is part of a European (especially French) tradition of courtly romance. It is more explicitly Christian than Beowulf, and is sometimes read as a Christian allegory.
In plot structure, Beowulf overcomes successively more powerful foes, while Sir Gawain is engaged on a single quest that has elements of moral test as much as tests of physical strength and warlike prowess. Both heroes confront their foes individually, with friends and companions in the background, but Sir Gawain is portrayed as one member of the court, a knight similar to the others, while Beowulf becomes the leader of his people.
When we read Beowulf, we see a ideal heroic character. He slays the evil creatures, he overcomes every obstacle in his life, he respects the community he lives in, also look for 'comitatus'. From the beginning to the end, Beowulf reflects the good side, aiding the people who need help (against Grendel especially) and receiving praise after every deed.
But in Sir Gawain and Green Knight, everything is not so bright about Sir Gawain, he does not prevail any deed unlike Beowulf. He actually 'fails' in his journey to Green Chapel. He betrays Lord Bertilak by not giving the girdle to him when they're exchanging items. Also, he loses faith by accepting the girdle, trying to elude his fate whereas Beowulf is deeply bounded with his religion. Sir Gawain is afraid of Green Knight all the time while Beowulf never scares of dying. In fact, dying is depicted as a 'resolution' in Beowulf, even Beowulf himself is aware of that every man has to die, so the goal of his life is to make best of it, this theme is also important in Homer's Iliad.
One last point, when Green Knight enters the Camelot court, and offers a challenge, nobody attempts to accept it even Sir Gawain. If he was a real true hero, he'd accept it before King Arthur unwillingly accepts it.
Still these are my comments, they could be wrong or not well explained.