How Massachusetts defines cider
The diminutive, angry state of Massachusetts has some decidedly bold feelings about what cider means... to them, at least. Via their official state website, they believe that "Fresh cider is raw apple juice that has not undergone a filtration process to remove coarse particles of pulp or sediment."
"To make fresh cider, apples are washed, cut and ground into a mash that is the consistency of applesauce. Layers of mash are wrapped in cloth, and put into wooded racks. A hydraulic press squeezes the layers, and the juice flows into refrigerated tanks. This juice is bottled as apple cider."
"Apple juice is juice that has been filtered to remove solids and pasteurized so that it will stay fresh longer. Vacuum sealing and additional filtering extend the shelf life of the juice."
The site also clarifies some of the danger (!) of cider, as this definition of cider (as it's often unpasteurized) is perishable, and will ferment if left unrefrigerated.
So, in essence, for the most part, apple cider is (basically) unfiltered apple juice. And -- due to the way it's produced -- will have a tangier, almost "harsher" taste than your traditional apple juice. This is probably why cider is oft-considered to be a more "grown-up" version of apple juice.