State nurseries (ie nurseries that are part of a school) will be staffed by qualified teachers and teaching assistants. It's rare for private nurseries to employ teachers because of salary costs.
I've had experience of both. My eldest two went to a private Montessori nursery and DC3 went to the nursery attached to our primary school (as will DC4).
Schol nursery is inflexible with times, you either do 5 morning or 5 afternoons. But loads of the private nurseries and childminders drop off and collect from school nursery. It's no different to after-school care, whereby the child goes to school and is collected by a childcare provider.
School nursery is very "school like" in my experience. It is still based around play, but everything has an educational context. I view school nursery to be more about kearning and nothing to do with childcare.
Because I don't work, my DD doesn't need childcare. But I would feel like I am denying my child an important educational experience if they didn't go to (school) nursery.
I would feel differently about private nursery though. If DD went to a private nursery, I would see that as childcare for MY benefit, so that I knew DD was looked after when I wasn't there. I would not view it as an experience that is important for DD.
So as a SAHM I wouldn't use a private nursery. I think DD would get more from spending her time with me. But because we have access to a school based nursery, I feel she gets a more valuable experience there than she would spending those 3h at home with me.
I guess that may all just be a reflection on the childcare providers around me, rather than a representative sample.