Another aspect of music is mental. I remember the first time I heard Liszt's Transcendental Etudes--there was so much going on that I couldn't process the music. I felt like my attention was a tiny pinpoint in the vast landscape of his music. I felt something similar with Bach fugues.
There are a lot of dimensions to piano playing. On the physical side, many of us could always move our fingers fast, but being able to control them and move them fast together and against each other takes much practice. For example, from doing exercises I can now raise and tap with any combination of fingers while leaving the other fingers relaxed--sometimes when I'm bored I do it on tabletops or desks. Friends who have never played the piano have tried to do it but couldn't--their fingers shake and twitch and jump, or it takes an ungainly amount of exertion to do it. And one of the goals of mastery is relaxation--where difficult things don't require the same exertion they would to someone who is not an expert at them. So part of progress is not just being able to play something, but to play it without tension. As you progress you'll probably revisit works you've learned before, and sometimes it is astonishing to come to a work that once seemed to demand everything from you, and now suddenly you can play it with grace and ease. Speed isn't just being able to move parts of your body fast, it's really about coordination and being able to align that with what you want the music to be.