A good fairytale will also show us something of ourselves/our human nature. That doesn't necessarily mean we will like what we see! The Little Match Girl by Hans Christen Andersen is, to my mind, rightly scathing of those who pitied the girl because she was dead but did nothing to stop her dying, which is the whole message of that story (and the exposure of hypocrisy).
A good fairytale will have memorable characters and there is usually a strong moral message with it (though conveyed in the story. A good story, of whatever type, will never leave you feeling as if you've been preached at).
A good fairytale will keep you gripped to the last world, will conjure up images of its setting and give you characters you can identify with/root for, even if they are strange alien monsters! A good fairytale will usually see injustices put right too.