I agree personal preference is a big factor. I like cannons because I like the subject of gunnery and the challenge to make cannons effective in ED's gamey combat model. I've tried frag cannons too but didn't like to have to get so very close to the target. I try to score hits with cannons at ranges of 1-2 secs flight time (less for small targets btw). Very satisfying to see a pirate in his gutsy FAS blow up 2 km away trying to run from me. With engineering It's easy to make every weapon OP against NPCs so I limit engineering to overcharged grade 1 that gives the same damage as premium synthesized munitions.
I don't mind the limited ammo. I don't like hanging out in RES or CZ for hours anyway. When the ammo is gone, I'm going home (and there's also synthesis).
If you are already flying to get close and get a "broadside" shot on your target, i.e. where the largest surface area of your target is exposed, to reduce the probability of a missed cannon shot then you are literally dogfighting in exactly the same fashion you would be when using frag cannons.
And in close quarters in your face combat where your goal is to expose the largest surface area of your opponent before pulling the trigger the frag cannons are far superior to the cannons anyways.
So why not just use frags in that case?
Because cannons have more range? Range means missed shots, so you close distance before pulling trigger anyhow.
I tried a fully cannon fitted and fully frag-cannon fitted Chieftain, and the frag cannon setup is vastly superior. It shreds most shields and absolutely wrecks hull. I've taken down everything from Sidewinders to Cutters with the frag-cannon setup, from fully shielded to zero hull, and the frags are simply superior in every way to cannons.
I honestly don't know why anyone would use cannons other than some personal preference for cannons and thus wanting to justify the use of cannons.
Low ammo, relatively low damage, high miss probability (especially on small targets) makes them undesirable.
And sure, some argue you can engineer the cannons. But that doesn't make cannons any better since, newflash, you can engineer frag cannons and other weapons as well.
Double-shot or High Capacity frag cannons with drag munitions, corrosive and incendiary shells will take down any ship much faster and more efficiently than cannons ever will.
I agree with Red Fox Four's points.
This debate echoes the age old "Regular cannon vs. LBX cannon" infighting debate in Battletech.
Given, however, that Elite is not constrained by the bounds of hexagon tabletop rules, and that there's been considerable time in game design to figure out possible answers to this...it's a bit annoying that it is echoed. Battletech is not well-known for being balanced.