hamlet acts more on impulse, horatio balances this out by thinking tghings through more logically, although they are both great thinkers
Hamlet says of Horatio "thou hast been as one, in suffering all, that suffers nothing, a man that Fortune's buffets and rewards hast ta'en with equal thanks; and blest are those whose blood and judgment are so well commeddled that they are not a pipe for Fortune's finger to sound what stop she please." But this is not a fair assessment: Horatio has hardly had "the motive and the cue for passion" that Hamlet has. Although he is fairly calm throughout the play, nothing is directed at him to disturb the calm.
Hamlet and Horatio only disagree once: over whether Hamlet was right to arrange for the deaths of Rosencrantz and Guildenstern. Horatio thinks (and most of us would be inclined to agree) that it is a harsh sentence to visit on two chumps who did not know that their commission would result in Hamlet's death (it was sealed remember) and who think they are only doing their patriotic duty to the King.
Horatio can see this because he is able to stand back from the action and observe it and judge it. Hamlet is, through no choice of his, in the thick of the action. He would love to be able to stand back and observe and judge it. He tries, but he cannot. He hates being "a pipe for Fortune's finger to sound what stop she please", or even a pipe that Guildenstern can play. But he cannot help it: when the circumstance is there he must act: he must stab the man behind the arras NOW without thinking; he must board the pirate ship NOW without thinking; he must jump into the grave NOW without thinking. Eventually he sees that "there is a special providence in the fall of a sparrow" and that it is futile for him to try to take control of his life, observe it, plan it, and execute what he has planned.
Hamlet recognizes that Horatio's status as an observer means that he must stay on to tell Hamlet's story. He is not part of the action as much as he might care for Hamlet. He is an onlooker.
But in essence the two men are not unlike which is why there is such an intuitive bond between them. What would Horatio have done, had his father been murdered, his mother married the murderer, his father's ghost come back calling for revenge and his girlfriend's father been a meddling spy who used everyone closest to him as a spy? Would he have done much different from what Hamlet did? Probably not.