Personal relates to you as an individual, not as a business person, public figure, etc.
Private means the information should not be shared.
For example, if I get a call at work asking me to pick up milk on my way home, it's personal (not work-related) but it's not private. It doesn't matter how many people know I need milk at home. If you spend too much time on the phone talking to family members, your boss may say "take care of your personal business when you're not on the clock."
On the other hand, my performance review or salary is work-related, not personal, but it's still private information not shared with others.
There is some overlap. If you asked me something about my sex life, I would say "Why would you ask me such a personal question?"