what is the use of the (hardback) tenon saw

A tenon saw is a type of backsaw, or a hand saw that makes use of a stiffened edge to give stability to the blade while it's cutting. The stiffened edge may be made of metal or wood and also give the blade weight. The tenon saw is the most commonly used of the backsaws.
Some types of tenon saw are designed to cut primarily with the grain; the teeth are rip filed, which works well for the type of work required to cut tenons. Others, which have crosscut filed teeth, can be used to cut against the grain. Tenon saws usually have between 10 and 13 teeth per 1 inch (2.54 cm) of blade. Most tenon saws have closed handles, although some smaller ones may have open or "pistol grip" handles.