Apple juice and apple cider come from the same process. Apple cider is just raw apple juice that has not been filtered, pasteurized, or processed. Apple juice is filtered to take out any impurities and only leave the pure juice. To read more about how the 2 are made, go to http://www.mass.gov/agr/massgrown/cider_juice_difference.htm.
The only difference between apple juice and apple cider is how it is made. Apple juice is made by the juice of the apples and then sugars and other items are added. Apple juice is also pasteurized. Apple cider is the whole apple and isn't pasteurized. It is not strained out like the apple juice is.
Apple cider has not undergone any filtration process and is considered raw apple juice while apple juice is juice from the apple fruits that has been filtered so all its solid parts have been removed. Learn more about these two type of juices by visiting http://www.mass.gov/agr/massgrown/cider_juice_difference.htm.