"Tangent's Cmoy" is just one layout of Chu Moy's circuit -- the only differences are just common ones like maybe I like to use bigger caps than in Chu's original schematic or something. It's not a significantly different design.
If you already have a layout (like my article gives you) duals are simpler -- only one chip to worry about, one set of voltage lines, etc. If you're starting from scratch, singles can sometimes be simpler to use depending on your protoboard's design -- their pins are less "dense", so you have more room to work with your wiring and external components. It's not so clear-cut an issue, really.
Personally, I only use singles when the chip in question comes in single-channel versions since in three different tests, I was unable to hear a difference between a pair of single-channel chips and the dual-channel version of the same design.
Duals are cheaper. Not only is one dual chip cheaper than two single chips, you have to buy twice as many sockets if you use singles, you may have to buy adapters, and you use more board space.
Some people will point out channel separation as an advantage of singles. Examine your op-amp's datasheets and decide for yourself. You'll probably find that any channel crossover ends up being inaudible at headphone amp signal levels.
One possible advantage of single-channel chips is that you can get them physically hotter than duals, which might matter if you're driving a chip right to its limits. That's not a situation you're going to encounter building a CMoy, though.