You may be wondering "If I'm basically just juicing apples, what's the difference between apple cider and apple juice?" Well, as it turns out, there's no widely agreed-upon distinction. I tend to think that cider is simply unfiltered and spiced, whereas juice is just the juice. Here are few other explanations, from The Straight Dope
"(1) There is no difference at all. (Source: large midwestern bottler.) Uncle Sam confirms that there is no legal distinction. In other words, it is all marketing booshwa. But see below."
"(2) The store-bought stuff is juice, the homemade stuff is cider. (Source: East Coast conglomerate; also, the old edition of the Encyclopaedia Britannica.) The product you buy from roadside stands usually has not been pasteurized. Consequently, it ferments over time, giving it a mildly alcoholic kick. What you buy in the store, in contrast, is pasteurized soon after crushing, preventing fermentation and resulting in a pleasant but kickless taste. The manufacturers call their product cider in the fall for marketing purposes."
"(3) Cider is made from apples that are picked early. (Source: Washington State outfit that claims to be the country's largest maker of juice and cider.) Early-harvest apples supposedly have higher acid and lower sugar content, producing a drink with a tangier taste. Thus true cider remains cider after processing because pasteurization doesn't affect the acid/sugar content. Therefore, the company claims, it's possible to make not only frozen cider concentrate, contrary to your assertion, but also "sludgy"--i.e., unfiltered, hence cloudy--apple juice. The guy I got all this from says his company is quite scrupulous about monitoring the acidity of its product and changing the labels accordingly."