Tossing lightly aside the chance to dress them up in pretty clothes, which is nice but easily fathomed, or the desire to write about social class or with historical accuracy, which is not what I'm interested in. . . .
Aristocrats have power. Clout. The ability to affect things. It is a lot easier to put an aristocrat beyond wondering where his next meal is coming from, or the long arm of the law. This means that his actions, suffering from fewer external constraints, can rove more freely, and are more driven by him. Which is much like giving him superpowers or magical abilities. By limiting the constraints, the character's free will gets much more play. A street urchin's decision to steal bread, or not steal bread, is probably heavily influence by both hunger and fear of the law. A nobleman who steals his neighbor's estate, or doesn't, is acting far more out of his own passions and will.
Indeed, there's a good chunk of things that are muchly faciliated with blue blood. Travel. Weaponry -- peasants may not have the right to carry weapons. Not wondering where your next meal is coming from. All of which can bog down a story not turning on them.
True, there are others who will likewise have powers. Where else could you get conflict from? But they will be limited in number and so a managable size cast.
But what when he is constrained? An impoverished patrician compelled to marry upstart money? Or dumped on the farm as a child and so raised in the belief that he was a farm boy? Made to work as a goose girl? Well, there, you have the matter of the escape. A born farm boy or goose girl who manages to escape is just lucky. One of noble blood has suffered an injustice. There's a great deal more drama where your situation has resulted from something positively evil rather than stemming from mere bad luck. And it adds punch to both the misery and the final happy ending to have them be injustice and vindication.