The flush cut saw has a thin flexible blade with teeth that have little to no set, and cuts on the pull stroke like a Japanese saw. It is designed to trim excess tenons, pegs and dowels so that they are flush with the surrounding surface without the saw blade digging in.
I use an inexpensive Irwin pull saw which costs around £10 and uses a replaceable blade, it’s mainly used for cutting off the wedges on Stable Stools like the photo below.