I had to reduce the original question because each sacred text mentioned could represent another answer. I think that reposting them into separate questions could be quite insightful. I think that one significant difference between both works is the relationship that individuals have to the divine. There is not really as close or tender of a relationship that Homer's characters have with the Greek gods as Arjuna has with Lord Krishna. In Homer's work, the gods are shown to be almost, if not more petty, than the human beings. This is not the case in the Mahabharata, where it is clear that Arjuna chooses Lord Krishna over all of the armies of the world and turns to him in a moment of agony. The entire song of the Bhagavad- Gita is one in which a mortal seeks the counsel of the divine and does the right thing because of the force of nurturing and care offered in the form of the divine. This does not exist in Homer's work. Hector is poised between equally desirable and ultimately incompatible courses of action. He is left to face his fate without the divine. Apollo does not help him. The walls of his city do not help him. Even his own family stays behind while he faces Achilles, only to be dragged and humiliated. This condition of being is not in the Mahabharata.
I think that this poses another area of contrast in both. The condition of forlornness is something that is intrinsic to Homer's conception of being a human. It seems that being alone and being placed in a condition where individuals must bear the curse of living alone is part of the Homeric struggle of being human. Achilles, though son of Zeus, essentially sojourns alone. Zeus does not give him much in way of guidance. Athena favors Odysseus and the Greeks, but she does not really present herself as one who is emotionally invested in the lives of her mortals. She helps them out, but it seems more out of pride or out of the desire to win that her motivation is forged. Divine figures like Ares simply enjoy seeing the humans scurry about and suffer. This condition of the divine leaves humans almost alone in the world. Priam begs Achilles for his son. He does not take solace in the gods. He accepts their will, but he begs another human being. In the Mahabharata, there is a reality offered that the pain and suffering of this life ("moksha") can be alleviated when we realize that Lord Krishna will never leave us if we take his name. The fact that Lord Krishna offers Duryodhna and Arjuna equal choice of he as a divine charioteer or his armies reflects that lack of forlornness in the human condition in the Mahabharatha. Lord Krishna already knows which one will choose what, but the fact that he offers himself to both the protagonist and antagonist of the struggle is a marked difference between both works. Lord Krishna demonstrates himself to be loyal to whomever calls out to him and waits for the right time to exact punishment to those who do not take his name. In this reciprocal condition, there is a difference in the relationship between the human beings and the divine that Homer seems to offer, where falling out of favor with will of the Olympians seems to change constantly.