I have three dc, all of whom have been in and out of both sectors and I don't see either as uniformly better or worse. It all depends on the school and the individual child.
Recently I decided to move one of my dc from an independent school with 10 in a class to one of our localish state schools which we both had a good feeling about and felt would be right for her. She did very well at the indie school academically and in sports but I moved her because of bullying and stealing and the oppressive atmosphere that resulted.
To cut a long story short, the head wanted to speak to me but didn't seem to want to discuss the bullying (specifically of a child with SN) and glossed over it. She then expressed concern that my child was now having to mix with the great unwashed(!)
I didn't realise indie schools were like this, or perhaps what I meant was I didn't realise this one was 
