Both the Wanderer (from the Anglo-Saxon poem "The Wanderer") and Sir Gawain (from the Arthurian text Sir Gawain and the Green Knight) are in search for something. Also, both men have been exiled (the Wanderer has self-exiled himself, and the promise Sir Gawain made forces him to exile himself as well). The Wanderer (who has lost his family, king, and kinsmen) is in search for "one in the meadhall who knew my people." The Wanderer is also in search for the one thing which will bring him solitude: his faith.
Like the Wanderer, Sir Gawain is in search of something. Having made a promise to the Green Knight, Sir Gawain must seek out the green chapel and receive a blow from the Green Knight. Similar to the Wanderer, Sir Gawain's faith proves to be very important (illustrated through his carrying of the shield which possesses an image of Mary).
Contrastingly, Sir Gawain does have family, kinsmen, and a king to return to. His loyalty to Arthur proves important (since he must not let down his king or bring dishonor upon him in any way). The Wanderer does not have to worry about this--his lord has been "laid in the darkness of the earth."
The characterizations of both the Wanderer and Sir Gawain prove that each man has love for their lord (both their king and God above). Each has a quest (although they differ greatly). Outside of these similarities, Gawain's quest will end with an embrace by his family, kinsmen, and king. The Wanderer, on the other hand, has no one. His quest will end when he meets his god.