If I had to compare and contrast Willy Loman from Death of a Salesman and Hamlet from Shakespeare, what would be their main similarities and differences?

Both Willy Loman and Hamlet are tragic and melancholic figures, but there is an important difference between the two: unlike Loman, Hamlet does not try to evade responsibility for his actions.
To forget his dismal failures, Loman tries to cocoon himself inside a strange, alternate reality. For his part, Hamlet simply accepts that the act of revenge will demand an unswerving commitment from him. True, Hamlet understands his dark destiny (and simultaneously loathes it). Yet, he understands that he must pursue a certain course of action if he is to avenge his father's death properly. On the other hand, Loman tries to evade personal responsibility for his actions, choosing instead to blame others for his failures and present unhappiness.
Let us first discuss Hamlet in act 5, scene 2:

Not a whit. We defy augury. There’s a special providence in the fall of a sparrow. If it be now, ’tis not to come. If it be not to come, it will be now. If it be not now, yet it will come—the readiness is all. Since no man of aught he leaves knows, what is ’t to leave betimes? Let be.

Here, Hamlet's words indicate that he believes in facing dangers with courage. For his part, Horatio urges prudence, but Hamlet is undeterred (despite his reservations). Prior to his fight with Laertes, we learn that Hamlet doctored the papers, so it is Rosencrantz and Guildenstern who go to their deaths. Here, we see Hamlet's decisiveness at play; he does not shrink from action when it is needed. Hamlet's decision to duel Laertes is also fortuitous because his subsequent wounding by Laertes provokes a confession. Before dying, Laertes reveals that Claudius was behind the plot to kill Hamlet.
Although he is dying, Hamlet seizes the poisoned sword and stabs Claudius. He even makes the king drink from the poisoned chalice (the same one Gertrude drank from). Claudius dies, but Hamlet's victory is a Pyrrhic one. Despite his fate, however, Hamlet remains resolutely in control of his mental faculties to the end, even designating Fortinbras as the next king of Denmark. Despite the fact that certain characters in the play have decried Hamlet's supposed "madness," the latter displays a surprising lucidity during the last moments of his death.
Hamlet's behavior is in contrast to that of Willy Loman's in Death of a Salesman. In fact, Loman never matches Hamlet's dynamism and thirst for action. To protect himself from pain, Loman relies on an alternative reality that allows him to forget his failures. He has an affair and deludes himself into thinking that his sons are interested in starting a business. To keep up appearances, Loman continues to take out loans from his friend, Charley. Here, you may be able to provide your own perspectives about Loman's behavior as well.
Although Loman is unhappy that he is not a successful salesman, he does almost nothing tangible to change his present situation. Although Hamlet and Loman are tragic characters, there is an important difference between the two. While Hamlet chooses to face the grimness of reality head on, Loman endeavors to hide from it. Loman's passivity is in direct contrast to Hamlet's ultimately aggressive approach to life's challenges.