The natural form of ZrO₂ is the mineral Baddeleyite. It has a Mohs hardness of 6.5. Obsidian is volcanic glass and typical glass has a hardness of about 5.5; typical window glass also has a hardness of about 5.5. Unlike ZrO₂ though, glass can be fashion to a blade easily and that would probably make it a better cutting tool.

I'm not entirely sure what are the properties of zirconium oxide, but I can tell you about obsidian. Obsidian, being a glass, has something called conchoidal fracture: when it breaks, it breaks into curved chips which come off of the parent rock learn more

Multiple fractures of these curved sections results in extremely sharp edges where the curved surfaces intersect, which is why it makes such a sharp blade. Of course, also because it is glass, obsidian is brittle, so for most purposes in which a good blade is needed, steel is more useful, because despite being much less sharp, it's also much less likely to break in the process of cutting something, because it's less brittle being a metal.

I hate to say this but define better. If you only care about which would make a the sharpest knife then obsidian wins because it can brake in such a way as to make very sharp edges. If you mean better as which will last longer and stay pointy longer, then zirconium oxide wins because it is not as easy to brake, does not ware down as fast, even though the edge might not be as sharp.

Obsidian: Very sharp edges, wares out quickly, breaks easily, can re make a sharp edge by breaking a new edge.

Zirconium oxide: Still darn sharp, edge lasts a long time, doesn't break as easily, doesn't need sharpening very often if ever.