1. Private lessons provide intelligent, instant feedback.
Once computers achieve sentience, online lessons may give traditional teachers a run for their money, but until then, the best that a program of online lessons can provide is information, video, and recordings. While these are great tools for enhancing one's knowledge and understanding of music, there is no substitute for a thoughtful and observant teacher who can catch problems as they are happening and correct them.
Music is best when played with appropriate expression. Good luck getting any feedback about this aspect from online lessons.
3. Customized Experience.
Teaching privately involves a lot of give and take. A great (or even a good) teacher can evaluate your current skill and knowledge and help to fill your most immediate needs immediately. Online lessons are not able to do this in any real way.
4. Personal attention.
The value of a working student/teacher relationship is one that is difficult to overstate. A full hour of personal attention every week from a teacher who has no other interest than to help you become the best at playing a musical instrument that you love has tremendous psychological benefits. Even a simple pat on the back for a job well done is something that online lessons can't duplicate.
5. Non-verbal transmission of skills and knowledge.
My most effective teaching occurs without speaking a word. There is no faster way to fix technical issues or to communicate interpretive subtleties than to show the student what they are currently doing, and then show them new possibilities for movement or expression. In fact, frequently, this is the only effective way to address musical and technical problems at the piano.
6. Creative give and take.
Playing piano is an interpretive art. It is not an exact science. If this were not the case, then there would be no reason to play the piano in the first place. What would be the point? Even the simplest of pieces offer many possibilities for interpretation. A teacher can help you decide which of your ideas are worth exploring, and which ones test the limits of good taste. Furthermore, a teacher can help you discover ideas that you would never have thought of on your own.
The value of private lessons far exceeds anything online lessons could hope to offer. There is no substitute for a great teacher.