A ban on the full veil in the United States would be unthinkable. This is in significant part owing to America's relatively small Muslim population, and its history of successfully assimilating generations of immigrants who have come to her shores determined to learn English and succeed on America's terms. But it is also because a ban would represent an unprecedented infringement of religious freedom. In America, the separation of church and state is designed to protect the state from religious interference and to protect religion, which always has a public component, from government interference.
France is different. It is home to approximately six million Muslims. That's more than in any other European state and represents almost 10% of France's population. Significant numbers of these relatively new immigrants are poor, confined to low-income and violence-prone neighborhoods on the outskirts of Paris, inclined to anti-Semitism, sympathetic to political Islam, and alienated from French social and political life.
In addition, the doctrine of laïcité—which is inscribed in Article 1 of the French Constitution and proclaims France a secular republic—separates church and state differently than in America. For many French, laïcité, roughly translated as national secularism, has acquired a militant meaning, according to which government must confine religion to the private sphere. This sensibility undergirded the ban on headscarves in schools.