For me the ability to go private just provides more choice of schools - you certainly can't guarantee that every private will be a better fit for your child than every state school.
I think the generic advantages that private schools have over state schools is that there is in general better facilities - none of the state primaries near me have any outside space that isn't concrete (even though it's a rural area) so no climbing trees, digging in mud etc. There are generally more opportunities for sport and music (funding cuts in the state sector have hit the local schools hard). A greater variety of extra curricular in general. Since they're not forced to follow the NC they can be a bit more flexible in what the students are taught so for example they can tailor the history curriculum to the interests of the particular class and they can spread out or accelerate the curriculum depending on the abilities of the individuals.
Mainly a convenience for parents but most private schools have flexible after and before school care - no waiting lists just ring up on the day etc. Again small numbers so kids feel really well looked after rather than being dumped in a big over crowded room.
There is less of a mix of demographics in the families. This is both an advantage and disadvantage. Most families value education and this is reflected in the fact that most kids want to do well. The disadvantage is that it's just a bit narrow - not everyone is super rich but everyone is comfortable and it's seen as normal that people have a few holidays every year and big gardens and nice birthday parties etc.
You also usually get smaller class sizes. You can't guarantee that private schools will have better teachers though and you have to be careful when comparing results as the kind of children at private school tend to get above average results where ever they go. They do tend to get more personal attention than most state primaries (even this isn't guaranteed some state primaries have small classes too).
The disadvantages are that children come from a wider area - you don't get the walking to school together, playing together in the same parks during the summer holidays. People might live a 40 minute drive away. Since it's a private business the parents see themselves as paying customers and can try to influence the school that way so I think school gate politics can be amplified. I definitely think the competitiveness can be amplified too since everyone can afford music lessons, tuition etc. If you're academic/sporty/musical you're likely to stand out less which can be a good or bad thing depending on the child. If there turns out to be need for specialist SEN support private schools aren't always equipped to provide it. (Most have an SEN for mild dyslexia etc but not a 1-1 TA or more specialist help).