There is no official difference between apple cider and apple juice in the US. In England cider is fermented and juice is not.
However it is usually true that cider is fresh pressed from apples while juice is made from concentrate. Cider is usually pasteurized but often unfiltered. Juice will be pasteurized and filtered as part of the concentration process.
There are ofcourse many exceptions to these generalities. Most apple juice in the US is concentrated so that it can more easily be stored for a longer time. The concentrate will last much longer without spoiling and takes up less room to store. This allows companies like Motts to have a regular supply of apple juice available to sell the consumer while apple cider is usually only available from local orchards in the fall and winter.
I think cider tastes much better than juice and would be my choice for making hard cider or cyser.
Craig