Cons:
* Poor firing rate
* Poor shot velocity
* Poor shot HP (projectiles are easily shot down)
* Recoil
Pros:
* High WMD (the highest, I believe, of any non-ammo-using weapon)
* Very high per-shot damage
* Fantastic device support (Makayev weapon enhancer; Lithium booster; Howitzer speed loader)
* High range
This makes it a poor weapon for fending off swarms of small ships, but a great weapon for dealing with capital ships and stations. As the game progresses, though, increasingly the threats the player faces come in the form of big, slow capital ships. There are still swarms, but they're relatively weak and usually serve primarily to drop the player's shields, not to scratch their hull.
Notably, howitzers are the only weapon type I know of that can get a stacking boost from secondary devices: +40% from the speed loader and +50% from the lithium booster. The speed loader boost actually is even better than that because it's multiplicative -- you get 1.4*1.5=2.1, instead of 1+(.4+.5)=1.9. And of course it gets even better if you apply tritium boosters -- no other weapon type can combine +fast with +150%.
In comparison, here's how I see the other primary damage types shaking out:
* (In the midgame) Particle weapons have fantastic fire rate, shot speed, and range, but have no WMD options and mediocre damage. Their device support provides a large straight damage boost.
* Ion weapons are basically up-statted laser weapons -- similar fire rate and range, maybe a bit more WMD, and of course more damage. The Tesla enhancer is seriously unappetizing as it only affects shield damage.
* Plasma damage is...I think I've tried one plasma weapon ever (in 1.7) and it was terrible? No device support at all, expensive, and a massive power draw makes experimenting with these unappealing. Plus there's not even any one-shot enhancers like the tritium injector or ion power capacitor.
* Positron: expensive, rare, and very power-thirsty. Has the same problems plasma does (the two are pretty indistinguishable in general).
Basically if you go with non-thermo for your primary weapon, then a) you're accepting lower DPS due to worse device support, and b) you have to rely on your launcher to provide WMD. Not the end of the world, but it does mean that your weapon isn't doing much to the compartments of all the capital ships and stations you're facing. Which, as mentioned earlier, comprise the bulk of the actual threats you're facing. Sure, you can take down swarms of Sandstorms faster, but you also have to spend more time getting pummeled by Deimos/Phobos lightning turrets and getting ionized by the Ventari.
A couple of special mentions:
* I think the Makayev enhancer also boosts the Makayev launcher, which is probably the best launcher for the midgame. Conveniently, it's also a fast-firing weapon useful for taking out swarms, so it complements howitzers nicely. Conversely, I'm pretty sure that the lithium booster doesn't boost thermo launchers.
* Howitzers are godawful at dealing with Luminous drones, especially since lategame launchers almost all use thermo damage which Luminous will rapidly learn to resist. The drones have a narrow profile and their shots will reliably cancel out any howitzer shots that were lined up properly. Fortunately Luminous isn't very aggressive and its stations have crap rewards, so if you're rocking the lategame howitzer you can mostly just ignore it.
* Of course if you're in a Wolfen then you probably won't want to spend your precious non-weapon device slots on enhancers.
If you disagree with me, hey, that's fine! But could you please share your favored lategame loadouts so I can try them out?

* The games where I don't use howitzers are usually when I decide to permit the thermo shell nanofac, which is even more broken than a twinked-out howitzer.