What are the differences between Ophelia's real madness and Hamlet's feigned madness?

In the simplest terms, Hamlet's madness comes and goes, Ophelia's does not. 

Hamlet tells Horatio in Act 1.5 that he is going "To put an antic disposition on."  In other words, he is going to pretend madness.  In Act 2.2, Hamlet tells Rosencrantz and Guildenstern that he is mad only on occasion, "I am but mad north-north-west."  Later, in Act 3.4, Hamlet tells his mother that he "essentially am not in madness / But mad in craft." 

Ophelia, on the other hand, tells no one that she is "mad"; she comes to see the Queen in Act 4.5 and her gentlewoman tells Gertrude that Ophelia is "importunate, indeed distract: / Her mood will needs be pitied."  The gentlewoman is saying that Ophelia is emotionally distraught and deserving of royal pity and attention because the Queen has said she will not speak with her.  When Ophelia comes into the room, she is singing snatches of songs. 

Later, Ophelia returns when Laertes is there and sings some more song snippets and hands out flowers and herbs.  Then, in Act 4.7, Laertes says to Claudius that he's lost his father to death and his sister to psychological desperation: "A sister driven into desperate terms." Toward the end of that scene, Gertrude tells Laertes of Ophelia's death, of how she fell into the water while trying to get some flowers and of how she seemed incapable of saving herself. 

In summary, Ophelia's madness seems complete because it is spontaneous and witnessed by others while Hamlet's is planned and questioned by others throughout the play.