My boys go to a private school in Devon and are very happy there. Until we moved, they went to their local primary school and were generally happy there too.
The main differences that I can see is that almost immediately at their new school, both of my son's were assessed academically and found to be off the scale in one area and reassuringly normal in all others. At their state primary, I was constantly being told that they were not quite keeping up in maths.
The truth of the matter is that they are completely on track for their age groups at maths and at least 5 years ahead in their literacy work. Private schools can support the whole child and schedule their work specifically for each child. The primary school was trying to get the boys maths up to the same level as their literacy to get amazing SATS results which was totally unrealistic and unfair. The children were very demotivated in maths as a result
Anyway, we had a choice of 2 local private schools and had 'taster' days at both. One was very grand in a faded gentry style. The other was more like a normal school in the juniors with the boys crossing the road at 11 to the grander senior school.
The big difference that I noticed was that the second school had far more older cars and wellie boots whilst the first school had lots of new cars and gucci boots at home-time. I had a good chat to someone who worked at the second school and she confirmed that many more working people used the second school whilst the first school was more old money-ish.
We have already had playdates over the summer with children from the school we chose (wellie boot not Gucci boot) and the kids have all been reassuringly normal!
We have an average income for our area but I am very good at budgeting, the fees for two are large but completely do-able. I would recommend having a savings account of at least a years fees before you start though.