The difference is similar to the way the health systems work – and, in fact, there is a large private sector in France but it is mostly integrated into the state framework. In the case of education the fees are heavily subsidised by the state in the vast majority of cases.
Subsidised schools are known as sous contrat meaning that they have a contract with the state and follow the usual national curriculum and take the standard exams.
About 17% of children go private in France and, of these, around 97% attend sous contrat schools (around 7% of pupils attend private schools in the UK).
In these schools teachers are paid by the state and the school receives money for its running from local authorities. They cannot discriminate on entry by faith, however the majority are run by the Catholic church and they are allowed to provide some religious education (the cost of which is excluded from the state grants). This means that fees are low – less than €1,000 a year and often variable according to means.
A few schools are hors contrat (not contracted). They have more freedom in the teaching and exams offered and are fully private in the British sense. These include ‘international schools’, where lessons can be in English and the exam sat is typically the International Baccalaureate.