What is the difference between infrared action spectroscopy and conventional IR spectroscopy?
After 30 minutes of Googling, I'm thinking that we both (EDIT: nope, it was just me) misparsed the name.

When reading the name "infrared action spectroscopy," my first instinct was to mentally split it up like 

$$\textrm{infrared action} \qquad\qquad \textrm{spectroscopy}$$

that is, spectroscopy of the "infrared action" flavor.

However, after running into more dead ends than a drunk rat in a maze with no exits, I realized that the correct way to split it might be 

$$\textrm{infrared} \qquad\qquad \textrm{action spectroscopy}$$

or "action spectroscopy" that happens to use infrared light.

Lo and behold, the first hit on google was a wikipedia page on action spectroscopy. That link seems to have more of a biological tint to it (as do most results), but there are several physical science papers out there as well. (See Edit)

It seems that action spectroscopy in general refers to hitting a system with light and watching some kinetic property of the system (rate of photosynthesis, conversion rate of isomers, etc.) In the case of the paper you cited, they report the relative intensity, so my guess would be that they did something like a scan over frequencies and watched how the ions fragmented at each frequency, which is supported by a cursory glance over the text surrounding Equation (1). Unfortunately, I don't have time to read the paper in full, so let me know if I'm totally wrong.

The moral of the story is that English is hard.

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EDIT: It turns out I'm the dolt and everyone else processed this correctly. As Brian mentions in the comments, the action spectroscopy page on Wikipedia is pretty awful, so I'll try to expand on action spectroscopy below, given what little I know about the subject.

In your usual spectroscopic setup, you hit the sample with some form of EM radiation, then look at what's reflected or absorbed to learn something about the system (e.g. UV-vis, IR, NMR, rotational)

It seems that in action spectroscopy, your goal is to use the light to cause some sort of a change in the system, and measure that instead. In the example given on Wikipedia, you measure the rate of photosynthesis as you change wavelengths--this tells you at which wavelengths photosynthesis is most effective.

In the paper I linked above, they seem to mediate the conversion of cis-butene to trans-butene using light. Presumably, you could start off with low-energy photons, then gradually ramp the energy up until you see the isomerization happen. The wavelength at which the transformations first start happening would tell you the activation energy barrier for the cis-trans isomerization. (Note that I can't actually read that paper because of the paywall, so I may be wrong about this.)

In the paper Brian linked, they seemed to use IR to study the fragmentation of an ion sample. They mention "loss paths" as observed through an MS, so it seems like they hit a sample with IR light, then figure out what it breaks up into.

As Brian said in a comment, it seems that action spectroscopy is where light is used "to alter the system under study, not to just probe it passively." I'd like to add that, in general, it seems that the quantity of interest is not the amount of light that comes back out, but something else that happens in the system as a result of the exposure to light (e.g. isomerization, ion fragmentation, glucose production).