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A stereotype is a trope gone bad.
The difference is subtle but telling. It's all about direction. The stereotype works backwards, grounding characteristics in nothing but the audience's assumptions. A trope--and a character--more forward.
A stereotype says, "Vampires are sophisticated and aloof; therefore, so-and-so is sophisticated and aloof."
A trope says, "So-and-so is a sophisticated and aloof vampire; therefore . . . "
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Stereotypes are lazy writing--the writer forces behavior for the sake of a plot (understandable; plots are difficult), then relies on stereotypes to excuse "huh, what?" behavior. Um, yes, I know I sent the vampire who doesn't want to be discovered to the school dance but that's um, because, because . . . he is a vampire-in-love!
Less lazy writers may still need to force the plot but they will delve into the trope to piece the characters' behavior together. Skilled writers will find a way to build off the trope, so the characters' behavior (the outcome) appears natural.