My experience has been that primary school can be very difficult for a high functioning autistic child which ever system you are in. We ended up patching together a mixture of homeschooling and 'alternative' schooling.
Private schools by their nature choose to cater to a narrow slice of the population, and if your child isn't their desired type they have no obligation to take them.
State schools have to take everyone, but resources are often stretched and some schools are badly designed for children who have sensory difficulties - too much open plan space, too much noise, too cramped, too much distraction in the classroom, too overwhelming. School lunch was a particular difficulty. There was also a lot of disruption to routine with things like World Book Day, Sports Week, World Week, Christmas Jumper Day, some random person popping in to talk about the Olympics Day (This was a while ago...). There was always a sense that we were putting the school out because DS didn't take these things in his stride like the other children and there was a lot of waiting for 'the call' to take him home because he wasn't coping.
Now that DS is at secondary school things are much, much easier. Partly that is the particular school he is at, but it's also that he is older and can articulate his feelings, so doesn't deal with a problem by running from the classroom, and many of the bits of school that he used to find difficult just aren't part of his day anymore. The UK system of allowing children to specialise early can be helpful for children who have 'spiky' abilities.
If you can afford to send your child to a private school you are lucky because you have a wider choice, but in the end it comes down to the school, the senco and your child's teacher, not whether the school is private or state.