Both young men have a similar job to do, but their personalities and their approaches could not be more different. Right away in Act 1 scene 1 we learn that Fortinbras has "sharked up a list of lawless resolutes" in order to attack Denmark and regain lands that were rightfully lost to King Hamlet in a battle some years ago. Fortinbras has hired mercenaries and gone behind his uncle's (the King) back in order to accomplish his goal. Notice that he has not enlisted the noble army of Norway, but instead a group of mercenaries that are in it for the gain and the glory. Fortinbras probably suspects that this time of transition in Denmark is a vulnerable one and sees this as the opportunity he has been waiting for. He is not specifically avenging a murder, but he wants to restore honor to the country and name of Norway.
Hamlet on the other hand is directly told to avenge his father's murder -- a murder that took place very recently. He is much more cautious and careful in his actions. He doesn't immediately act on the information from the ghost, but goes about to test the truth of it by pretending to be crazy and by putting on a play that suggests the actions of Claudius, wherein, Hamlet can judge Claudius's reaction to the play and determine his guilt. It is only after he has proof that he can act, and even then, other obstacles "inform against him." He chooses not to kill Claudius during his prayers and misses an excellent opportunity; he accidentally kills Polonius, making himself a more imediate crisis to Claudius; hence, he is sent to England by Claudius. Hamlet figures out a way home and does eventually deal with Claudius, but it is too late to save himself, and he dies from the poison that Claudius and Laertes use in their plot against him.
All that said, it isn't that Hamlet doesn't do anything -- it is just that he doesn't do it with the same kind of intense and focused action that Fortinbras does. If he had acted more aggressively and earlier, he may have been able to avoid his fate. It is a nice irony that in the end, Fortinbras is able to "with sorrow embrace [his] fortune, " and he can take not only the lands his father lost, but the total throne of Denmark!