One important difference between sodium and ammonium thiosulfate, is that at the same pH, ammonium thiosulfate has a greater reducer effect on the silver image than sodium thiosulfate (Russell & Crabtree 1932). Hence the recommendation not to over-fix when using rapid fix.
That's why Haist recommends a sodium thiosulfate fixer for the "finely divided silver image" of certain papers (591). If you're using Azo, for example, I suppose you should stick to traditional fixer.
Ammonium thiosulfate is also relatively less sensitive to the buildup of iodide ions than sodium thiosulfate (K. Frank 1966). Because modern film emulsions contain a certain amount of silver iodide, you're better off with a rapid fixer, acid or alkaline (Haist, 589).
I'm curious to know if anyone has compared the look of prints fixed with either rapid or traditional fixer. My guess is that the difference is perhaps too subtle to be noticed by the casual observer, and that the recommendation not to over-fix is really just to ensure that people don't destroy their prints by letting them in the rapid fixer for an entire printing session. But I'd like to know.
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H.D. Russell and J. I. Crabtree, "The reducing action of fixing baths on the silver image" J. SMPE 18: 371 (1932).
K. Frank, "Some aspects of developer and fixing bath concentrates based on potassium and ammonium salts respectively," J. Phot. Sci. 14 (1): 44 (1966).
Grant Haist. Modern Photographic Processing. Volume 1. John Wiley & Sons. (1979)