"Spirulina" is NOT Spirulina
"Spirulina" is Arthrospira.
Spirulina is a genus of cyanobacteria that is not "Spirulina".
Got it? Good.
The confusion lies in the fact that one is a genus of cyanobacteria which forms spiraling chains of cells and the other is an entirely different genus of cyanobacteria which forms spiraling chains of cells. Oh, and both belong to the same order: Oscillatoriales.
Health food companies, not caring either way, sell Arthrospira as "Spirulina". Some sources attempt to clear up the confusion by putting one of the genus names in parethesis or listing both genus names before the species (for example, Arthrospira Spirulina platensis). Some scientific papers which discuss "Spirulina" will note that the genus is actually Arthrospira, then use the "Spirulina" name for the rest of the paper.
What in the world is the difference? To quote SPIRULINA (ARTHROSPIRA): AN EDIBLE MICROORGANISM. A REVIEW., "Spirulina and Arthrospira morphologies are differentiated fundamentally by: helix type, distribution of pores in the cell wall, visibility of septos under light microscopy, diameter and fragmentation type of Trichomes (filaments) (Guglielmi, et al., 1993; Vonshak and Tomaselli, 2000)."
Check out http://www.google.com/search?hl=en&lr=&ie=UTF-8&safe=off&q=spirulina+arthrospira to see the taxonomic mess for yourself.