The diagnostic feature is gill placement. Sharks carry their gill slits on the sides of the head; rays, including sawfishes, carry them underneath, and the pectoral fins are broadly attached to the head. Sawfishes belong to the order Rhinopristiformes among the batoids, and are distinct from sawsharks, which are true sharks with gill slits on the sides and a pair of barbels on the rostrum.
The saw itself is a working instrument. High-speed video of related species shows sawfish slashing sideways through schools of fish to injure prey, and the rostrum is studded with electroreceptors used to detect animals buried in sediment.
The saw is also the reason the animals are nearly gone. It entangles instantly in nets, so sawfish are caught by almost any coastal fishing gear, and mangrove nurseries have been extensively cleared. Every sawfish species is now Critically Endangered, and the US population was the first marine fish listed under the Endangered Species Act.