When talking about the sharpness measured in atomic widths, you have to look at the molecular structure.
Diamonds are cubic. This means two things
1. They have distinct 90 degree cleavage (look it up if you think breasts. Check out the geology related cleavage article if you want to know more) in all 3 dimensions. This means that the edge will be flat because a large number of layers will just slog off of the cutting edge.(if you have a big diamond in your/your spouse's ring, you will want to be careful not to knock it on anything hard. It's easier to break in two than a piece of glass the same size. If the atoms are jolted just slightly out of place, the atomic repulsion will do the rest and your diamond will cleave.)
2. The two sides of the blade will always want to be 90 degrees from each other, but you may be able to grind it down to around 45. The repulsion of each carbon atom to each next will keep you from getting a stable edge.
Obsidian is a glass. Glasses being amorphous, do not have any cleavage so easy to shatter along. It does however have conchoidal cleavage, though its comparatively weak.
This actually works to a glass knifes advantage. If you take some obsidian and knap off two sides, you can get the two sides to be around 33 degrees or even less depending on how fast the glass cooled and what minerals it is made up of. The concoidal cleavage means a smooth cutting surface, and the amorphous, but bonded structure, means that you can get down to a molecule thick. These blades are actually MORE durable AND SHARPER than a diamond blade. Try it. Break a peace of glass and see how easy it is for that glass to cut you. Some man made glass is specifically designed NOT to have these edges, so get a non shatter proof glass. Im not liable when you get cut or worse, you were warned.
Obsidian, specifically, has been used for a very long time in the medical profession and was the prized tip for spears. We have had the sharpest knives on earth since before we could write. Now days they are used in ocular surgeries when lasers are not available. As far as the famed diamond blades are concerned, its their HARDNESS (mineralogical term. look up Mohs hardness scale) that gives them their notoriety. If your mineral is harder on Mohs scale than mine, you can cut mine. If mine is harder, your just going to blunt your edge. Now days, we harvest non jewler grade diamonds and artificial diamonds to COAT blades with. To cut glass, jewels or just anything you want a precise cut on, you use diamond because it will keep its edge the longest, seeing as its the hardest. There's also nothing it cant cut, while glass which can be sharper, is only a 5.5 on this scale to 10 (diamond). Glass will loose its edge on about anything.
It'd make a wicked blade till you hit something other than flesh though!