The Hanging Cheek / Filet Baucher is a leverage bit which places pressure on the horses poll when the reins are applied encouraging the horse to lower its head and is thus suitable for horses with a high head carriage. The top, smaller rings are attached to the bridles check pieces with the larger bottom rings being attached to the reins of the bridle.
The Elliptical Link Snaffle Bit has a rounded central piece and therefore gentler in the horses mouth. A popular bit across multiple disciplines as it is easier accepted by most horses.
The elliptical mouth reduces pressure on the tongue and bars of the mouth as well as the palate. Elliptical mouths encourage a softer contact.