Both Hamlet and Fortinbras have to avenge their fathers. Examine their approach to this task. What similarities and differences do you see?

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!