The short answer is: no, they are not the same, but they are terrifyingly similar—like fraternal twins who went to different finishing schools. Granulated sugar is your standard, all-purpose sugar: medium-sized crystals that are sturdy and reliable. Caster sugar (also called superfine sugar in the U.S.) is its more delicate, powdered-in-secret cousin with much tinier crystals.
Think of it this way: if granulated sugar were a gravel truck, caster sugar would be fine beach sand. One is built for roughing it in the baking wilderness; the other is for sophisticated, airy pastries that whisper sweet nothings to your taste buds. Size really does matter here, folks.