Several differences. Legally, foglights are illegal to have switched on if it is not foggy (or very poor visibility) - darkness doesn't count, whereas normal dipped head lights are legal. Foglights must be mounted below the front bumper. Foglights are, by definition, blinding as their lenses break the light differently than headlights. Often, they are yellow since this penetrates fog better.
Normal heqadlights are placed in such a position that their beam is reflected back into the driver's vision by the fog, causing dazzle. Fog lights have a flat topped beam and are placed lower down, so that their beam is not reflected back into the driver's face. They are not a perfect solution, but help. Use them with a dipped headlight beam, of course.