Clive Wearing, the subject of the article, is able to play and recall pieces of music he knew years ago, even though he is not capable of remembering what he did a few seconds ago. He is able sightread at the piano or the organ, speak and write in several languages, dance, and perform hands-on tasks like making coffee and dressing himself elegantly, because he somehow physically understands what to do.
I have always had a hard time memorizing music. If I were pressed to learn to play a piece from memory, I don't think I could do it, yet if I were to have a modern flute, an instrument I haven't practiced in twenty years, in my hands, I know that I would be able to play a series of about ten orchestral excerpts, a bunch of etudes, and scales, and a bunch of repertoire, including pieces that I won't remember until I have the instrument in my hands and I am prompted by the first two or three notes.
If I have a fiddle in my hands, I can play parts of a handful of pieces that I know from muscle memory and by ear due to 15 years of practice: a page or so of solo Bach, a page or so of a Mozart concerto; but if I were to try to play the Beethoven Sonata I am working on right now (that I practiced carefully not an hour ago), the only way I could call it into my mind would be by visualizing the music on the page. The hard parts--the parts that are hard to both decipher and to physically play--won't make it into my visual memory for at least a month, and I know that I will never be able to play it from memory (though my goal is to be able to simply play the piece).
I can play many of the Bach cello suites by ear on the viola because I used to hear them practiced every day when I was a child. When trying to actually play them from memory, unless I am not paying attention at all and happen to make it further, I pretty much crash and burn after a page or so. Playing a whole suite or even a whole movement of solo Bach from memory is something that I know I will never do. It's a good thing that I have the music.