Hamlet becomes melancholy and depressed after his father's death. He is so overwhelmed that he goes into a long period of mourning. He essentially withdraws himself and acts strangely. His lingering sad and depressed state provokes the concern of both his mother Gertrude and his stepfather (and now-King) Claudius, who express the same in Act l, Scene 2:
Good Hamlet, cast thy nighted colour off,
And let thine eye look like a friend on Denmark.
Do not for ever with thy vailed lids
Seek for thy noble father in the dust:
Thou know'st 'tis common; all that lives must die,
Passing through nature to eternity.
'Tis sweet and commendable in your nature, Hamlet,
To give these mourning duties to your father:
But, you must know, your father lost a father;
That father lost, lost his, and the survivor bound
In filial obligation for some term
To do obsequious sorrow: but to persever
In obstinate condolement is a course
Of impious stubbornness; 'tis unmanly grief
Hamlet resents their advice because he feels they do not share his sorrow and pain. He is upset that his mother married his uncle so soon after his father's death, and suspects that they had an adulterous affair before his father's demise. He sees their relationship as incestuous and has very little regard for either of them.
Furthermore, Gertrude and Claudius's marriage denies Hamlet the throne. Claudius is the new liege, and Hamlet despises this fact. He does not see Claudius as his father's equal. He is heartbroken and, at the end of his soliloquy in the same scene, acknowledges that he should refrain from expressing his disdain:
But break, my heart; for I must hold my tongue.
When Hamlet later learns of the true nature of his father's untimely demise from its ghost, he is sworn to vengeance. The ghost informs him that he had been murdered by Claudius. Hamlet does not act immediately, however, and delays his revenge. He does concede, though, that his indecision is borne of cowardice, and he hates himself for procrastinating. In the end, though, he formulates a plan to exact his revenge.
Laertes's response to his father's death is more immediate and passionate. Once Laertes receives notification of Polonius's passing, he immediately returns from France and demands to see the king. He is intent on taking revenge and initially believes Claudius is responsible. Once Laertes confronts the king, he demands to know how Polonius died, saying,
How came he dead? I'll not be juggled with:
To hell, allegiance! vows, to the blackest devil!
Conscience and grace, to the profoundest pit!
I dare damnation. To this point I stand,
That both the worlds I give to negligence,
Let come what comes; only I'll be revenged
Most thoroughly for my father.
Laertes's fury is worsened when he sees the state of his sister, Ophelia, who seems to have lost her mind. He becomes easy to manipulate by Claudius, who uses the young prince's passion to plot Hamlet's death. Laertes, intent to be appeased, agrees to be directed by him as long he is the one who kills the prince:
My lord, I will be ruled;
The rather, if you could devise it so
That I might be the organ.
One must also understand that both characters discovered the true nature of their fathers' deaths differently. Hamlet initially believed his father died of natural causes, and only later did he learn from his father's spirit that Claudius had murdered him. Laertes knew from the outset that his father had been killed.
Furthermore, Hamlet had to contend with the veracity of the spirit with which he spoke. There was no direct proof that Claudius had killed King Hamlet, and the prince was in a quandary about the nature of the ghost. Laertes had to deal with none of these issues. He knew Hamlet killed his father, and his only consideration, therefore, was quick revenge. In contrast, Hamlet, deep thinker that he was, had to figure out many permutations before acting.