Both Gilgamesh and Arthur are kings of their respective realms. Arthur was king of Britain and Gilgamesh, of Uruk. Although Arthur comes to his kingship by pulling a sword out of the stone after being guided by Merlin, he is still considered to be a fully human character by most accounts. Gilgamesh, on the other hand, is two-thirds god and one-third man. Although Gilgamesh is part-god, his partial humanity means that he, too, will suffer a mortal death.
While both King Arthur and Gilgamesh are both called great kings, they have very different ways of ruling. Gilgamesh was considered more fearful than honorable, and he had very little respect for the "merely human" people who were his subjects. In fact, before his counterpart and friend Enkadu is sent by the gods to distract Gilgamesh, Gilgamesh regularly slaughtered and terrorized his own people. King Arthur, in some accounts, is depicted as a godly, Christian king. Since there are many King Arthur myths, though, there are many different ways of seeing his character. Arthur is also depicted as irresponsible and even child-like, taking most of his advice from Merlin, and unable to function without Merlin's help. While both kings have weaknesses, Gilgamesh seems far more callous about the lives of his subjects.
In both epics, a quest is a central part of the story. In Gilgamesh, the king undertakes a journey to meet Utnapishtim, a person who is immortal, in order to get the secrets to immortal life. Gilgamesh begins this quest after the death of his only friend, Enkadu, when he realizes that he, too, will die. In the King Arthur myth, Arthur and his Knights of the Round Table see the Holy Grail in a vision. After the vision, many of Arthur's Knights go after the grail in order to bring it back to Arthur's kingdom of Camelot. In some versions of this story, Arthur is actually displeased with their quest, as he fears that they will all die in trying to attain the grail. While the grail quest is a clear part of Arthur's story, it differs from Gilgamesh's quest in that Arthur himself does not undertake the quest.
Both of these tales end in the death of the hero. Gilgamesh returns to Uruk with the knowledge that men are mortal and that he cannot escape his fate. This seems to make him a wiser, more compassionate ruler, and when he dies, his death is mourned by the people of Uruk. King Arthur dies in a battle with his own son, Mordred. Arthur's death is much more ambiguous than Gilgamesh's. In some legends, Arthur dies and is buried as any other king might be, while others claim that Arthur is taken to Avalon, where he will eventually return and reclaim the throne of Britain in the future.