I've had children at both state and private schools. My youngest 2 aged 7 and 13 are both currently privately educated.
From extensive experience I would say that the private school wins hands down for me - if you can afford it I'd definitely recommend it.
The prep school is like being part of an extended family, although the one mine go to is much larger than yours. The class sizes are still small though - around 13-15 children in each.
There are just SO many reasons for me, I'd be writing a huge post. I'm really happy for you to private message me with any questions though but I don't want to bore people.
You mentioned activities and this is a huge part of school life. There are just so many things to choose from and mine have activities every single day after school and occasionally before. In fact they even shoot and fence on Saturday morning. The school has fantastic facilities, including 2 swimming pools as well.
Also although my children at state primaries were very happy theres no doubt, parents were not really present much. We'd wave goodbye in the playground in the morning, attend an assembly once a term and perhaps a sports day and collect from the playground in the afternoon. In the lower prep school though, you drop your children right to the classroom (which has windows in the wall to see inside from the corridor) and when they leave the teacher shakes each of their hands individually. The children say "Good afternoon Mrs X" before leaving. Its nice waiting for them in the warm and I loved seeing them get organised for the day after they'd left me. In the upper prep the teacher brings the class to the main hall and you meet them there after the goodbyes. You are allowed to watch every single after school activity and many parents turn up for these - you feel connected to the school in a way that I never did with state. That's not a criticism at all, the state primaries do a marvellous job with huge classes and stretched budgets. The wrap around care is great for working parents too, but for SAHM like me, it still gives great peace of mind to know that if you ever can't get back in time you can just call and send them to after school care.
The medical care is second to none - they employ their own full time nurse and the sick room has a couple of beds in a beautifully decorated room. The children who fall over etc and end up there usually get cookies and milk!
The teachers take lots of photographs of everything and these are emailed home (you can of course withdraw consent for this but I don't know anyone who has).
With the senior school there's less parental involvement and it's not relevant to you at this stage but I'm happy to discuss this with you too, but once again it's very different.
I'm not really doing it justice - but if money isn't an issue I'd wholeheartedly recommend it.
Please do contact me if there's anything specific you want to know - I could talk forever on this subject. I think my 7 year old has a magical time at school. I wish it was me :)