How are Horatio and Hamlet different?

Though Hamlet and Horatio are both scholars and friends who become very close over the course of the play, the two men are, as your question points out, markedly different. One particular area where this difference is most apparent is in each man's willingness to accept the supernatural. Specifically: Horatio is a skeptic and Hamlet is a believer.

Horatio's skepticism is one of the first things we learn about him. As Marcellus brings the reluctant Horatio along to meet with Barnardo and watch for the ghost, Marcellus says Horatio "will not let belief take hold of him." In fact, Horatio is so sure the ghost is a "fantasy" that he scolds the two men, saying, "Tush, tush, 'twill not appear." (I.i.23-30) 

Compare this reticence with Hamlet's response to hearing about the ghost of his father. Whereas Horatio had to be dragged out to watch for it, Hamlet is quick to say "I will watch tonight. Perchance 'twill walk again." (I.ii.242-3) Hamlet even goes on to talk about the possible portents of this ghost's appearance. 

Even after he's seen the ghost twice, Horatio is hesitant to accept what he has seen. As the ghost's voice encourages Hamlet's friends to swear to keep silent, Horatio cries out in disbelief, "but this is wondrous strange!" Hamlet, however, has no fear of the strange. He plays on the use of the word as he replies, "And therefore as a stranger give it welcome." Hamlet then goes on to paint the clearest difference between their two world-views: "There are more things in heaven and earth, Horatio, then are dreamt of in your philosophy." (I.v.164-7) Horatio tends to believe only what is in his books; Hamlet is willing to accept a wider world. 

Horatio himself paints one last clear distinction between them at the very end of the play. As Hamlet is dying, the prince asks his friend to tell the truth of what has happened. Horatio responds that he won't go on living because he is "more an antique Roman than a Dane." In other words, though Hamlet has acted the part of the brave Dane in fighting for his vengeance, Horatio has more in common with Roman soldiers who were known to commit suicide and die beside their slain commanders.

Hamlet stops him from killing himself, however, and after Hamlet dies Horatio speaks uncharacteristically poetically about his friend and the events that have occurred. So perhaps a bit of Hamlet has rubbed off on the skeptical scholar.