In all of Shakespeare's plays, there are only three true "fools": Feste in Twelfth Night, Touchstone in As You Like It, and the Fool in King Lear. These are characters who are employed by a personage of high standing—like a Lady (Feste), Duke (Touchstone), or King (the Fool)—not only to entertain them but also to "speak truth to power." Shakespeare's fools are thoughtful, witty, well-spoken, and invariably the smartest person in the room.
Shakespeare's "clowns," on the other hand, are simple and often simple-minded characters, not particularly well-spoken, and "a little rough around the edges." Clowns aren't particularly astute or perceptive, and they only occasionally, sometimes purely accidentally, speak truth to power. Shakespeare's clowns are the funniest person in the room but not the smartest.
Shakespeare's fools are self-aware. They know they're smart, and they know their worth to their masters. Shakespeare's clowns aren't self-aware but are, more often than not, blissfully unaware of the world around them and of the importance of any insightful remarks they make. Clowns might think they're smart, or they might have an inflated opinion of themselves and their worth to society, but they're simply fooling themselves.
Shakespeare's fools add intellectual and philosophical depth to his plays, whereas his clowns simply add comedy.
It's not a coincidence that Twelfth Night, As You Like It, and King Lear were written in 1600 or later, which coincides with the time that Robert Armin (c. 1563–1615) joined Shakespeare's acting company, the Lord Chamberlain's Men, replacing Will Kempe (c. 1560–1603) as the lead comic actor. Will Kempe was essentially a rough-edged, purely comic actor who played characters like Peter in Romeo and Juliet, Costard in Love's Labor's Lost, Lancelot Gobbo in The Merchant of Venice, Bottom in A Midsummer Night's Dream, and possibly Falstaff.
Some scholars believe that Shakespeare replaced Kempe with Armin because Kempe had too great a tendency to overact and improvise his lines beyond what Shakespeare had written for Kempe's character. (Hamlet's "advice to the Players" in act 3, scene 2 of Hamlet might well have been directed at Kempe.) Robert Armin was a more refined comedian who could sing and dance very well, as Feste is required to do in Twelfth Night. Armin played more intellectual, philosophical, and "wordly-wise" characters than did Kempe, but Armin could also play the rougher comic characters like the Porter in Macbeth.
Dogberry, the "master constable" in Shakespeare's Much Ado About Nothing, is often referred to as a "fool," but he's not. Dogberry is foolish, yes, but Dogberry is a clown. Dogberry is a role for Will Kempe, not for Robert Armin.
Nevertheless, Dogberry occasionally reveals incontrovertible truths in Much Ado About Nothing through his malapropism-filled dialogue.
DOGBERRY. Why then, depart in peace and let the child wake her with crying; for the ewe that will not hear her lamb when it baes will never answer a calf when he bleats. (3.3.63-65)
Dogberry also uncovers...
Don John's plot against Don Pedro, Claudio, and Hero, but he and his fellow constables are too bumbling and inept to reveal the truth about Don John's plot to Leonato in time to prevent Hero's utter disgrace at her wedding. Dogberry's discovery of Don John's evil plot and his eventual revelation of the plot to Leonato have a significant effect on the play, but this result is purely accidental.
All the while, Dogberry is much more concerned with the fact that Conrade called him an "ass."
DOGBERRY. (to Leonato) Moreover, my lord, which indeed is not under white and black, this plaintiff here, the offender, did call me ass. I beseech you let it be remembered in his punishment. (5.1.296-298)