Concerning spying in Shakespeare's Hamlet, I'll focus on a different angle than the answer above. Both Polonius and Claudius are characterized by their actions.
Polonius is revealed as hypocritical and petty and as an interfering parent. He has barely finished preaching to Laertes and telling him to be true to himself, when he arranges to send someone to spy on him. Based on everything we see of him in the play, Laertes is a model son. He is even respectful to an elder that is extremely hard to be respectful to--his father, Polonius. Gertrude isn't. She loses patience with his hyperbole. The act of spying on his son is hypocritical and petty, and Polonius has no business doing it.
Claudius is revealed to be insidious and underhanded. When he orders the interchangeable Ros. and Guil. to spy on Hamlet he may remind one of Macbeth in the tragedy named after him, when Macbeth says that he has spies in the houses of all of his thanes. He may be wondering if Hamlet is "mad," but this is secondary to his wanting to keep an eye on Hamlet in order to protect himself.
Claudius, though highly intelligent and capable, also makes a mistake here by underestimating Hamlet. Hamlet sees through his plan, humiliates the two spies, and reveals to Claudius only what he wants to reveal to Claudius.
Polonius, you will remember, is all art, and no matter. In Act II, scene I, Polonius spends about 50 lines trying to justify his methods of spying on his own son. He could have said it in only 10, which is how he ends the exchange with Reynaldo:
See you now;
Your bait of falsehood takes this carp of truth:
And thus do we of wisdom and of reach,
With windlasses and with assays of bias,
By indirections find directions out:
So by my former lecture and advice,
Shall you my son. You have me, have you not?
By contrast, Claudius is to the point. And in Act II, scene ii, he says to Rosencrantz and Guildenstern:
I entreat you both,
That, being of so young days brought up with him,
And sith so neighbour'd to his youth and havior,
That you vouchsafe your rest here in our court
Some little time: so by your companies
To draw him on to pleasures, and to gather,
So much as from occasion you may glean,
Whether aught, to us unknown, afflicts him thus,
That, open'd, lies within our remedy.
Granted, a King may be more brief than an advisor. And a King may not be questioned, (the way Reynaldo attempts to question Polonius), for that which comes from a King's mouth is gospel.
The other difference is that Polonius' spying is unwarranted and a systematic pattern of paranoid behavior. Not only will he spy on his own son, but he will spy on his potential son-in-law, Hamlet. And he will sacrifice his two children in doing so, much like Claudius sacrifices his two pawns, Rosencrantz and Guildenstern. So says Enotes editor:
Polonius’s tedious instructions to Reynaldo echo his earlier advice to both Ophelia and Laertes, and foreshadow his behavior in Scene 2. When Reynaldo asks why he is being sent to spy on Laertes, Polonius’s justification is so circuitous even he loses track of what he was saying: “What was I about to say? By the mass, I was about to say something! Where did I leave?” (49-50).
I think the major similiarity, of course, is that both Claudius and Polonius are asking people to spy on other people to gather information about them. As you say, Reynaldo is to spy on Laertes and the other two are to spy on Hamlet.
The main difference, to me, is that Claudius has more of a reason to want to spy on Hamlet. Polonius is only spying on Laertes because he feels like it. He has no reason to believe that Laertes is going to do anything bad. By contrast, Claudius has plenty of reasons to believe Hamlet is crazy. Asking the two men to find out why he is crazy seems to make a great deal of sense.