Guillotine may be messy, but the head is severed before the nerve endings can send pain signals to the brain.
To a culture squeamish about the gas chamber or hanging, the exsanguination of the guillotine would provide plenty of horrific ammunition to outlaw the death penalty altogether. You can argue the medical realities all you want, but image is everything, and the vision of heads rolling while decapitated torsos gush blood is one to haunt all but the shrieking hordes at the Place de la Concorde.