Of course, the real reason they use dilithium is that TOS's writers made up (di)lithium crystals as a plot McGuffin without bothering with the technical details, and later creators had to work with what they were given.
This is very true, and always important to bear in mind.
During TNG, Sternbach, Okuda et al stated (unofficially... ie, in the TNG TM) that dilithium is used to "moderate" the reaction... which really isn't quite the right term (since "moderate" usually means "help bring together" in this sense... and instead, the dilithium is really helping to keep them apart. Or rather... it ensures that the reaction is "on particle at a time" or something to that effect... a relatively small, continuous stream of energy instead of a single massive BLAST.
I don't personally prefer that explanation, by the way... I'm just restating one of the more popular "conceits" about what Dilithium does.
In my "personal canon," Dilithium is something entirely different. I base this upon my own personal interpretation of everything I've seen in TOS and TAS, and less on what came along afterwards. But it fits everything seen in TOS and TAS, and isn't directly contradictory with anything seen on-screen in TNG or other TNG-era shows (although it is contradictory to what's in the TNG TM).
See, for me, I look at the TOS crystal, which was a flat "slab" of crystal with a pair of terminal knobs cut into the crystal at either end. This sort of configuration for materials is not unheard of in real electronics design... and was more common at the time of TOS (when all components were "discrete" and the concept of the "integrated circuit" was still just a concept, not reality).
The "dilithium crystal" seen in TOS looks very much like a photodiode might.
A photodiode is the exact opposite of an L.E.D. With an L.E.D, you put voltage in, and get light out. With a photodiode, you put light in and get voltage out.
When I see this TOS crystal, I see a device that, if subjected to certain types of high-energy radiation (and we know pretty well what EM output a matter/antimatter reaction produces, because we've experimentally determined the answer), produces a high output voltage.
We know (from "sparks and electrical shorts" and so forth) that the TOS Enterprise uses, at least in SOME situations, and frankly, it seems to be in virtually ALL situations, "simple electricity" as the primary internal power source.
This is not unreasonable. Electrical power distribution is a pretty fundamental "laws of nature" thing... used by nature, used in our brains and bodies, used in all of our technology. There is no REASONABLE argument by which you could conclude that "they won't use electricity in the future" which wouldn't be a clear, "bad science fiction conceit."
SO... to me... dilithium crystals are the means by which the high-energy reaction products from the matter/antimatter annihilation reaction are converted into usable energy (in the form of some type of electrical power).