King Hamlet and Prince Hamlet are foils, so there are more differences than similarities. Many interpret King Hamlet to be the vengeful God "Yahweh" from the Old Testament and Prince Hamlet as the meek Jesus character from the New.
Prince Hamlet is unsure the Ghost was indeed his father or a demon. So, he stages a reenactment of murder, the Mousetrap, to test Claudius' guilt and the Ghost's story. Only after he sees Claudius' reaction does Hamlet believe the Ghost.
King Hamlet killed Fortinbras Sr., which shows his cruelty. He also plunders after battle. This shows he is a faulty leader, as faulty as Claudius, whereas Hamlet and even Fortinbras Jr. are more contemplative philosopher-kings.
As for a similarities, it may be said that both father and son loved Gertrude, a kind of Oedipal love triangle. Although, the Prince thinks she is complicit in King Hamlet's murder. He is cruel to his mother, and King Hamlet's Ghost appears when Hamlet is about to go too far. The Ghost reminds Hamlet to "leave her to heaven."
Both are loyal: King Hamlet to his wife, even though she married his brother; Prince Hamlet to his father, even though he commands him to kill the king.
Both the Ghost and Hamlet enjoy theatricality: the Ghost's monologues are the best in the play, and Hamlet is a captive audience. Later, Hamlet stages a play for Claudius, reenacting the Ghost's words. So, Prince Hamlet directs the script given him by King Hamlet's Ghost.
Both King Hamlet and Prince Hamlet are caught in Purgatory. The Ghost literally is caught there, and Hamlet is placed there by the Ghost's demand for revenge, which goes against the Prince's nature.
What other critics say:
Kenneth Gross claims that the Ghost "is, like Hamlet, a figure at once subjected by and giving utterance to slander and rumor.” (60) Both the King and Prince act on supernatural slander and rumor, as there is never really proof the Claudius murdered his brother.