Catherine & Charlotte seen on The Tube
So if they can travel on it without fuss, private citizens can
Hazno and the ILBW confuse (they confuse a lot of things, as they are not the brightest, aren't they?) security with privacy. Catherine and Charlotte were safe - they most likely had hidden security with them - but people recognise them and took the video, so they couldn't just have a private day out.
If you are a public figure, you have to accept that people will recognise you, take your pictures. It's not the same as having to accept to be harassed.
What it shows is that if Catherine and Charlotte, the future Queen and Princess Royal, can take the tube, a traitor of an ex-prince, his Z-list actress wife and their back-of-the-heads-children are perfect safe stuck in traffic on a school run.
It also shows why everybody loves Catherine and loads the ILBW. Catherine accepts the extraordinary position her marriage provides but understands her roots and the privilege of simple, ordinary life. That's why she is one of the most privilege women in the world and still most people relate and empathise with her.
The ILBW doesn't understand her privilege as she only sees hierarchy and she craves to be on the absolute top. So she is always trying to project something she is not, dreaming of a delusional lifestyle she feels entitled to. That's why her popularity is still going down.