Apple Cider has more of the fruit in it, so I think it tastes more like apples and I guess it’s more nutritious, too. It is usually pasteurized and refrigerated. It is popular to add spices to cider, but the gallon jugs you see at the store are just pure juice, unless otherwise labelled.
Sparkling Cider is more like bubbly apple juice sold as an alcohol-free champagne substitute.
Hard Cider is an alcoholic beverage… I’ve never had it, but I doubt that the mass-produced stuff has much apples in it.
Apple Juice is more processed and filtered, so it is transparent and doesn’t require refrigeration prior to opening the bottle.