For a lot of us (and I am include myself to some extent in this) school was difficult. At best the odd ones out - at worst horribly bullied (and often in a "psychological" way so whispering/rumours/always being the butt of jokes. That stays with you. I work in a professional job and have a lovely group of friends (including several school Mums), but I still occasionally get that tight chest feeling when at a children's party/school run when someone says something. (Fortunately, my children have been ok on the friendship front and no issues of bullying; however, a lot of their friendships have "thrived" because we are close friends with parents).
I'm in the SE and lots of us are not from the town we live in. Lots of people don't have local family, they move out of London when they have a baby and so lots of people make friends at baby groups and through children (I have some amazing friends from different groups). I can imagine if you are new, a bit lonely, and go into a playground which seems really clique it would be hard, while someone just saying hi, asking how you've been can make a difference. I often go to coffee with some Mums after the school run on a Friday and I always feel better for having a good chat and putting the world to rights.
Those bullies from childhood rarely change - you see them in the playground or the whatsapp (guarantee those who've been bullied can pick them a mile off). . As I've got older I recognise often the snide comments in fact hide their own insecurities and can ignore, but it still not pleasant. For some people that cliqueness will spill over into their children then not being invited to things.
For me its not wanting approval, but it's more a "for goodness sake why do you have to be so unpleasant". As I say I am fortunate I have made some lovely friends on the playground (think people you know you call at 3 am and ask to drive you to the hospital, people who checked in and helped when my DF was dying), the people when you have a horrendous morning give you a hug and some kind words, and so I get why if you don't have that, you stand their on your own, the school run could be a difficult experience. Some people won't be bothered - but that doesn't mean that people who are are odd/strange or a bit pathetic.