Fire blankets are used where extinguishers are not suitable for the given risk. Typically they are used in commercial and domestic kitchens to tackle cooking oil fires. If a fire extinguisher is aimed into a burning pan the sudden expansion of gas as it is forced under the surface of the oil could cause the oil to burst out of the pan, spreading the fire. A fire blanket on the other hand smothers the fire, starving it of oxygen.
Safety note: A pan which has been involved in such a fire must never be moved until it has cooled down, which will be much longer than you might think.
Fire blankets in larger sizes are also used where people work near flammable liquids. A victim whose clothes were alight would be laid on the ground and the blanket used to wrap around them, again smothering the fire.
(See also Class F Extinguishers, which have been developed specifically to tackle the risk from modern high temperature cooking oils.)