The nature of the reactions, despite 'application of energy', is quite different; the energy involved is also on vastly different scales.
Cauterization involves application of heat, burning the tissues (killing the tissues) and denaturating the proteins (completely wrecking their structure), causing them to precipitate out of solution and clump together, plugging things up (plugging up bleeding blood vessels, and also causing blood to clot around the plugs, being a side effect of it). Lots of heat energy, sufficient to burn flesh. Usually done with a fair bit more precision these days of course.
This technique, on the other hand, is quite similar to one which I use from time to time for disinfection of periodontal pockets around teeth... application of a dye (in my case, toluidine blue) which binds to the bacteria, and then activation of the dye with the appropriate frequency of light which is matched to the absorption spectrum of the dye (sorry, not at the office so can't look up the specs), generating free radicals which react with bacterial components and ultimately killing the bacteria. The energy involved is literally that of the photon of the proper wavelength which knocks the electron out of the dye when the dye absorbs it... multiplied many times of course. Without the matched dye to absorb the light though, the light won't be doing a heck of a lot of useful work... sure it'll be absorbed by other molecules, which does heat them up somewhat, but nowhere near the level of heat used in cauterization.
In this situation, the dye is Rose Bengal, which likely has an affinity for collagen. Activation of the dye causes the collagen molecules to form bonds with one another, cross-linking them. Essentially, it turns the existing collagen where it is applied into the 'glue' to hold the wound together. No destruction of living tissue as cautery would (whether tissues die from other factors with the injury, such as insufficient blood supply, are a different story), and also much less of a mess of various byproducts left behind afterwards as well.