Our ancestors discovered that certain stones were very hard and, when broken, they formed very sharp edges. These were logically used to produce cutting tools such as knife blades, arrowheads, axe heads and scrapers. Geologists have refined classification of these various stones into categories such as flints, cherts, jaspers, chalcedonies, agates, quartz, obsidian, etc. For the purposes of simplicity it is easy to lump them into two broad categories: Flint-like and obsidian.
Flint-like stones all have a microcrystalline structure that breaks with a conchoidal fracture. They are very hard (about a 7 on the Mohs Scale). Their hardness and fracture qualities make them ideal for creating a durable cutting edge. If you need a durable cutting tool, a flint-like stone would be your best bet.
Obsidian is natural glass that was made by volcanic action. It also breaks in a conchoidal fracture and has a hardness of about 5.5 on the Mohs scale. Since it was formed by quickly cooling, it does not have a crystalline structure. Obsidian fractures to form an edge even sharper than the flint-like stones. Fractured obsidian can form an edge sharper than high-quality steel scalpels. Since it is essentially glass it is more fragile than the flint-like stones. If you needed to create a scalpel, then obsidian would be your best bet. That’s not to say that you can’t still make a fine tool from it. It is just softer and more brittle than the flint-like stones. Obsidian can also be polished to form a mirror.