In what ways is Rama different from Gilgamesh? Think about this in terms of a hero. Do you think of him as a hero?

The nature of their heroic quests is a point of differentiation between Gilgamesh and Lord Rama.
The heroic quest for Gilgamesh takes on different forms. Gilgamesh is searching for self-discovery.  He is "a man of many moods'' as both person and political leader.  He shows restlessness through his self-indulgence as a king. His own people pray to alleviate them of the pain from his rule:  "You made him, O Aruru, now create his equal; let it be as like him as his own reflection, his second self, stormy heart for stormy heart. Let them content together and leave Uruk in quiet."  Upon leaving Uruk, Gilgamesh changes as a result of friendship with his kindred spirit, Enkidu.  With Enkidu's assistance and encouragement, Gilgamesh kills Humbaba.  Slaying this adversary is more for his own legend than anything related to his people. Gilgamesh's heroic quest takes on a different form when Enkidu dies. Gilgamesh strives to find answers to issues of human existence such as life, death, and the desire to achieve immortality.  While Gilgamesh grasps how "there is no permanence," most of his heroic voyage is done to find some level of immortality.  In his return as a wiser and more thoughtful king, the belief is that Gilgamesh has found immortality in his journey and its implications.  His heroic journey is steeped in gaining insight for himself.  He voyages to find truths to questions that plague him and, while he does return as a better ruler, the heroic quest was mostly for his benefit and understanding.
Lord Rama's heroic journey is constructed in a much different way.  Whereas Gilgamesh searched for understanding and meaning, Lord Rama clearly understood his purpose.  An an avatar of Vishnu, Lord Rama wedded himself to dharma, or maintaining the cosmic order of the universe through rightful actions. While Lord Rama never carried himself as divine, being an avatar of Vishnu makes him different than Gilgamesh's two thirds divine, one third human.  Lord Rama's embodiment of duty based on dharma endeared himself to the subjects of his kingdom, Ayodhya.  They loved Rama and valued him as their prospective king.
Lord Rama is exiled because of his adherence to dharma. Unlike Gilgamesh who chooses to go outside of his kingdom, Lord Rama is banished from his. When Kaikeyi insists that King Dasaratha exile his beloved son, Rama does not object because he sees his dharma as adhering to his parents' wishes. Unlike Gilgamesh's subjects who yearn for their leader to leave, the citizens of Ayodhya weep when Rama, Sita, and Lakshmana leave, following them into the forest to sleep at their beloved prince's feet.  
At no point in his heroic journey does Lord Rama question his purpose.  He knows that his entire being is dedicated to dharma.  He commits himself to representing it in his actions and thoughts.  When Lord Rama slays demons, he does not do so to add to his personal legend.  He does it because of his obligation to help those in need.  When he kills Ravana, he does so because of the existential threat posed to justice.  He knows that Ravana threatened the order of the universe. Evidence of this lies in how Rama did penance for killing Ravana, something that Gilgamesh would have never done.
One final difference between both heroes is in the truth that Gilgamesh discovers.  The reality of impermanence haunts Gilgamesh.  What he does both before and after this realization is meant to offset the crushing inevitability of time.  He wishes to establish permanence in an impermanent world.  In contrast, Lord Rama is not really concerned with impermanence. One reason might be because he knows that dharma is a part of universal reality.  By wedding himself to duty, Lord Rama knows that he is permanent because duty, itself, is permanent.   Lord Rama's heroic voyage is thus taken for the benefit of others and not for himself. 
Both Lord Rama and Gilgamesh can be seen as heroes. They are heroic in what they did and heroes in the truths they represented. If there is a differentiation between them, it might lie in the realization of purpose.  I think that Lord Rama's heroic journey is more focused and purpose-driven, something that Gilgamesh's journey lacked at different points.  The reader has to develop their own metrics in determining heroic value as there is ample evidence to suggest that each is a hero.