Is there a difference between Hertz and 'frames per second'? It's not uncommon that the term 'frames per second' (sometimes abbreviated as fps or FPS) is associated with, or even equated to, the unit Hertz (Hz). I'm not exactly sure how these two concepts relate to each other.
The difference I can see is that SI defines Hertz as $\text{s}^{-1}$, which makes it a unit, whereas 'frames per second' includes both a physical phenomenon (frames) and a unit ($\text{s}^{-1}$), which makes it... something inbetween a unit and a quantity?
To which (ontological) category does 'frames per second' belong, if any? If it belongs to the category of units, is there a difference between 'frames per second' and Hertz?
 A: "Frames" is not a unit of anything.  A frame is a thing.  FPS in Hertz measures frames in one second.
More generally Hertz can be used as the unit of any "thing" per second.  In the case of an oscillating wave we measure cycles per second in Hertz.  When I was young, there was no "Hertz", and the units were "cps" and "fps".   Those old designations were more explicit, and avoided the confusion you are having.
A: Hertz should be understood to mean "periodic events per second". I your case the events are the display of frames, so yes, you would be perfectly justified in using $\mathrm{Hz}$.
That said, as several commenters have already mentioned, the unit "Hertz" does not specify what kind of periodic behavior is being counted. So the author(s) or speaker must make that information explicit in some other way. The use of "frame per second" is one way to accomplish that: it is a more specific way to express yourself. 
The real stinker with Hertz comes when you want to compare two closely related quantities like "angular frequency" and "frequency". The former is generally expressed in radians per second and the latter in revolutions or cycles per second. Transcribed into pure SI units both get written as $\mathrm{Hz}$ but they should not be treated as equivalent to one another. As a stop gap measure you can keep symbols for radians ($\mathrm{rad}$) and revolutions ($\mathrm{rev}$) around as if they were units, but they both represent pure numbers.

In the question text you actually pose a linguistic or philosophical question 

To which (ontological) category does 'frames per second' belong, if any?

As I understand it, SI simply views this as the counting of something that happens periodically and as such "frames" is just a number without dimensionality.
There is no fundamental reason it must be that way, but the committee has decided to duck the question of what countable things should get their own units by saying, in effect, "none of them". Anything that you can count (in whole or fractional numbers) that happens periodically can have its rate denominated in Hertz.

An aside is that $\mathrm{Hz}$ is not the only SI unit equivalent to $\mathrm{s}^{-1}$. The becquerel ($\mathrm{Bq}$) also has that dimensionality, but should be understood to mean the (average) rate of some random or probabilistic behavior (such a radioactive decay).
A: A frame is one image (produced by some imaging device such as a computer monitor). Frames per second is therefore the measure of how many unique images are produced in one second (i.e. the frequency of the frames). Hertz is the SI unit of frequency, typically used as a measure of cycles per second. When you're referring to cycles or frames, you're not talking about some quantity such as mass or length, but how many frames or cycles there are (i.e. the count of frames or cycles). The "unit" for counting things, isn't anything, so it is sufficient to quantify Hz as a unit to measure of the frequency of "things".
A: Units are arbitrary, for instance, some systems use the value $c=1$ for the speed of light.
"Frame" is a unit, but because it is not a SI one we can set it to "vanish" without affecting any other units. Something similar happen with radians. An angular speed of 1/s might mean 1 rad/s or 1 turn/second, depending on context.
