Do Claudius & Polonius have the same reasons for spying on Hamlet & Ophelia? How are their goals similar in Hamlet? How are they different?

In Shakespeare's Hamlet, Claudius and Polonius, on the surface, have the same reasons for spying on Hamlet and Ophelia, but only on the surface. 

Both Claudius and Polonius are trying to figure out why Hamlet is "mad," or acting insanely.  Polonius believes that Hamlet's behavior is due to his love for Ophelia being rejected by Ophelia (which he put her up to in the first place).  He is trying to prove his theory to Claudius.

Claudius wants to know if this is the reason for Hamlet's insanity, too.  In this respect, Claudius and Polonius are similar.  But Claudius has a further motive, as well.  He is trying to keep an eye on Hamlet and figure out Hamlet's behavior, so he can judge whether or not he is in danger from Hamlet. 

Claudius, of course, has recently kept Hamlet from inheriting the thrown by marrying Gertrude, and has also "replaced" King Hamlet as Hamlet's father, at least technically.  Neither of these make Hamlet happy.  Claudius wants to know if Hamlet is so upset due to Ophelia's rejection, or due to the other recent events.  Claudius is hoping that Ophelia's rejection is the cause of Hamlet's mad behavior.  If that is the case, then Hamlet is no threat to him. 

Of course, according to Hamlet himself, his behavior is all an act, a smoke screen, if you will.  Hamlet says that he is only pretending to be mad, in order to, we assume, keep Claudius busy worrying about his madness, rather than about Hamlet's plans for revenge.  Hamlet's madness is a diversion.