To be alone would be that you are alone in place, people would leave you alone with your thoughts for instance.
To be alone and private - would be that you want to have a moment of solitude, reverie, you want to have some contemplation time without people interfering with your thoughts.
By being alone we can understand that one could be alone and private, but not always being alone and private would mean the same as alone.
Let´s say, Robison Crusoe being a shipwreck survivor (at the time he thought he thought there wasn´t a soul in the island) he was technically alone (what could be a moment of solitude, contemplation, a time of being only with your personal feelings, but not a moment of being "alone and private".
To be alone and private would suggest that he wished to go to the place and that he would think that it was going to be salutary, a healthy thing to his mental health. That is more often than not the way one thinks when faced with the expression "to be alone and private".
So, not always to be alone means to be "alone and private", the latter can be (it usually is) a thing longed for, the former is not always wished or healthy.