Beowulf and Sir Gawain are characters from different stories, but they share many similarities. Before we dive in, keep in mind that each character is truly a reflection of the cultures in which they were created, so we're dealing with the Middle Ages.
One rather stark difference between Sir Gawain and Beowulf concerns religion, a very important value in the Anglo-Saxon medieval era. Beowulf often prays to a Christian god but does not seem particularly religious, whereas Sir Gawain is very devout, wearing the symbols of Christianity.
You may know that the medieval period was quite fixated on bravery. It's not surprising, then, that both Beowulf and Sir Gawain have distinct interactions with courage. Beowulf is endlessly brave, never fearful of battle or the monsters who threaten him. Sir Gawain, on the other hand, exhibits a lack of courage when he battles the Green Knight. This failure to show his bravery leads to his realization that courage is an essential value to a truly honorable knight.
We also see the idea of quests and adventures manifest in these characters' stories. Beowulf travels to the land of the Danes to face Grendel and prove that he is a notable warrior. Gawain, after he accepts the Green Knight's challenge, travels to Green Chapel to face the Green Knight once again. Both characters leave their homes to prove themselves as warriors.