A major difference between Stavrogin's and Ivan Karamazov's "faith" is that Stavrogin begins with ill faith and later looses it, and Ivan, on the other hand, lives in poor faith but never looses it. After Stavrogin's search for God turned to failure he lost all hope for any kind-of faith, because he has even lost the hope of believing in Him for reasons sake alone. In a conversation with Shatov, he reminds Stavrogin of his own words, "To cook a hare - you must first catch it, to believe in God - you must have God" (p 259). Stavrogin always lacked true faith.
Ivan Karamazov hated God, but he believed in Him. He couldn't forgive God for all the evil that is going on in the world, and he even went so far as to create a file against Him. Ivan believed that people are inherently evil, "The beast of rage lays in every man"(p 222).
Both Ivan Karamazov and Stavrogin don't know how to treat God, and what to take Him as. Stavrogin wants God the person or object, and Ivan wants God the law and facts. They both miss the point that God is neither of those things and appears through faith and love.
Stavrogin in his search for God becomes a god himself - a kind-of idol, to himself and others around him. He had to commit suicide because no human can be god. The human mind is too meek, too insignificant to comprehend the secrets of the universe. Stavrogin was too deep in and could not handle it anymore. Instead of fighting the pain or finding salvation in Dasha he decided to end his pathetic life by killing himself in a very undignified, almost humiliating way.
The reason that Ivan Karamazov "gets away with a mere nervous breakdown" and a discussion with the devil is because he is a Karamazov, and as a Karamazov he has a positive (one of very few) strength - love for life. This love for life is Ivan's salvation. Through this love he also loves humanity and therefore is capable of finding Christ in his life.
Bibliography
The Brothers Karamazov, Fyodor Dostoevsky (Modern Library College Edition); Translation by Constance Garnett
The Devils, Fyodor Dostoevsky (Penguin Classics 1971); Translation by David Magarshack.