Beowulf is a traditional Anglo-Saxon epic and Sir Gawain is part of a later romance tradition. This means that the two epics are very different in style, tone, and attitude, despite both having noble heroes who are models of virtue as well as physical prowess.
Beowulf is written in Old English alliterative verse, bearing a strong affinity to the Norse sagas. The hero is distinguished by physical strength and nobility, and part of a system of hereditary affiliations and reciprocal debts. The ethos is an admixture of pagan and Christian, and fame, as it reflects upon one's lineage, is central to how a hero lives on after death, rather than the narrative having a purely Christian sense of the afterlife.
Sir Gawain is embedded within a Christian and courtly tradition. Women play a much greater role in this epic, and how a hero treats women is considered a measure of virtue. The hero faces a moral test based on resisting sexual temptation and dishonesty rather than just tests of physical strength and valor. Unlike the monsters of Beowulf, the Green Knight is not evil, but in many ways a mentor who helps Gawain grow in virtue.
Similarities: They are both adventure stories where a hero accepts a challenge, sets off on a quest, and for the most part succeeds in that quest. They are both about legendary characters who are serving their kings. They both face dangers and foes along the way but survive and come out triumphant. They both succomb to temptation from the ladies. They both receive laud and honor for their deeds. The stories themselves are both oral traditions that were eventually written down, surviving the ages. They both have great feast-halls with mighty kings. Both stories have Christian overtones.
Differences: Beowulf succombs to vice on a much larger scale than Gawain. The foe Beowulf faces is an evil creature, whereas Gawain faces a mythical, noble knight that teaches him a valuable lesson. Beowulf's deeds save many people and resuscitates a kingdom whereas Gawain's quest is more to defend his and his king's honor; it's not such a dramatic situation.
Those are just a few ideas to get you started, and I provided links to more thorough discussions on both stories. I hope that helps!