How are the heroes Sir Gawain (in Sir Gawain and the Green Knight) and Beowulf (in Beowulf) similar and different?

In his own way, each hero represents the values of his
culture.  Beowulf represents the warrior colde of the Anglo Saxons:  bravery, loyalty,
physical strength.  Gawain represents the chivalric code of the medieval period which
also involves courage, loyalty, and physical prowess.  However, the two heroes differ in
the way that they demonstrate these virtues.  Beowulf meets evil monsters which
represent the antithesis of the warrior code:  Grendel, Grendel's mother, and the fire
dragon.  Sir Gawain's test is not one against evil; it is one that tests his own
integrity.  Sir Gawain's opponents, in other words, are not evil antagonists.  They
provide instead an opportunity for Sir Gawain to prove his character.  Beowulf can
defeat the evil monsters, but Sir Gawain is unable to demonstrate that he is perfect. 
Gawain comes close to perfection, but fails by taking the green sash offered to him by
the lady of the castle. 

At the end of the story, even
though Gawain has performed bravely and done more than any other knight was willing to
do, he felt himself a failure.  Gawain's adventure demonstrates the medieval idea that
man should strive for perfection even if he falls short.  Beowulf, however, is
victorious.  He is able to defeat the monster and save his people, even if at the end he
loses his life.  In the Anglo-Saxon epic, the threat comes mainly from without.  In the
medieval romance, the threat is actually a test, not truly a threat at
all.