It can come with some slight personality changes or quirks that would make the character behave more like the original story aspect, but nothing like a full possession or change to the character. So for example, if a character has an aspect that is anger management issues like the Beast from Beauty and the Beast, they won't suddenly be belligerent. However, they might get a little annoyed more easily at certain things, and if they do get mad then it might explode much more than they normally would.

Another example would be having the Mad Hatter and wanting to throw tea parties or drink tea a lot more often.

Hope that helps!