Throughout this web site we refer to modeling in 1:64th scale as "S-scale". You will find magazines and other web sites using the term "S gauge". That terminology is not correct, but it has its roots in the early 1900s when the hobby of model railroading started. You will find similar terminology used in other scales as well (e.g. "N gauge", "HO gauge").
The word "gauge" refers to the distance between the rails, whereas "scale" is the ratio of prototype (real-world) size to the modeled size. As a side note, the word "gage" means "pledge". So, "gauge" should be used to represent the distance between the rails, and the device by which such distance is measured.
There are two common uses of the word gauge that we use in model railroading, namely, "standard-gauge" (4'8-1/2" of spacing between the rails) and "narrow-gauge" (e.g. 3 feet between the rails), yet locomotives that run on these rails are both scaled to 1/64th of the real thing.
Due to S-scale's history being tied to American Flyer, there is further confusion in that, within the S-scale community, there are sub-divisions, such as American Flyer, hi-rail, scale, narrow-gauge, etc. Some refer to "S gauge" as meaning either American Flyer and/or hi-rail, and "S scale" as being scale modeling (standard-gauge or narrow-gauge). This is incorrect, because all models are still 1/64th of the real thing, so all are indeed S-scale.
If the distinction needs to be made between "American Flyer", "hi-rail", or "scale" modeling, it will be so indicated on this site using those words as shown in these quotes. The NMRA standard uses the term "deep-flange", which would apply to S-scale's American Flyer and hi-rail modeling (anything that uses or requires code 125 or taller rail).
Within S-scale, the difference between all of these sub-divisions really only boils down to whether or not your equipment uses deep flanges and/or American Flyer-compatible claw couplers, versus more to-scale flanges and the smaller couplers. Other distinctions, such as level of detail on the models, and whether or not animated accessories are used, are independent of these definitions and are really a personal decision the modeler makes to enjoy his or her involvement in the hobby.
Past NASG presidents Roy Hoffman and Bill Winans have publicly expressed that we should start to use the term "S-scale" instead of "S gauge". Also, past NASG Vice-President Dave Heine wrote an article in the February 2009 issue of the NASG Dispatch (pg 20), explaining the difference between scale and gauge.