What specific sugars are found in apple cider, apple juice, and apple cider vinegar?
There certainly are chemical differences between apples and commercially purchased apple juice, but it's impossible to tell what they are, they are trade secrets. 

Apple juice bought in stores is not "apples squeezed then bottled", such a juice would become unappetizing before it was transported out of the factory. Juices are high tech products - "100% apple" just means that all ingredients were derived from apples, but the producer can have changed them in whatever ways it saw fit, before making a final product with the right consistency and aroma. 

If you are talking about freshly squeezed apple juice at home, the only chemical difference between that and whole apples is the oxidation of some antioxidants in the apple, mostly polyphenols. The sugars are not changed chemically.