As far as I'm aware, the ballots aren't secret from the people doing the voting - the burning, etc, prevents them from being reused and the voting being tampered with.
From an MSNBC article, regarding the Room of Tears, "They call it the “Room of Tears” because several times newly elected popes have entered there and broken down in tears, realizing the responsibility that they have and the enormous burden they have just taken on to represent God on earth."
It's an unofficial title, but I suppose if enough guys break down in there, it might gain a reputation
