Ah, that.
The first thing to know is that the range cap on non-missile weapons means that you're never going to get to utilize the extra range on higher-tier particle beams and lasers. The second is that non-turreted weapons can only traverse at the ship's maximum speed. That aside:
Lasers: Long range, damage which falls off gradually over range. Damage profile is a single column of armor, as deep as the hit is strong. In other words, a Str-9 laser hit would burn through 9 points of armor in a straight line from the outside of the hull towards the interior. What this means functionally is that lasers are exceptionally good if yours are strong enough to punch through enemy armor, but largely worthless against large ships with armor heavy enough to preclude one-shot penetration (because of the lower odds of a second beam landing on the same armor column as the first). Can be turreted or spinal-mount in addition to normal fixed-point hull mounts. Are effective PD weapons, especially smaller ones, and (IIRC) they're the best against plasma torpedoes. Has reduced-size/increased-recharge-time variant tech which makes them suitable for fighters.
Railguns: Somewhat shorter range than lasers, but no damage falloff. Damage profile is akin to buckshot, a number of smaller "craters" taken out of the surface, not dissimilar to missile impact patterns. Thus, railguns are the opposite of lasers in terms of strengths and weaknesses vs. armor. Cannot be turreted.
Particle cannons: Longest theoretical range of any E/KW, no damage falloff, but not particularly spectacular damage either. I don't even remember what the hell their damage profile looks like because they're almost never used except very occasionally by Precursors. Cannot be turreted.
Gauss cannons: Extremely short range, only deal a single point of damage. Almost useless offensively since they sandpaper off single tiles of armor, though they can work in a pinch. Excellent PD weapons when set to final defensive fire. Can be turreted. Suitable for fighters.
Meson cannons: Medium range, deal a single point of damage. They ignore shields and armor to strike directly at the components of enemy ships, and are the only E/KW which can fire through atmosphere without penalty. Obviously they're good both for taking down difficult-to-damage targets and for planetary PD installations on colonies with Earth-normal atmosphere. Can be turreted.
Plasma carronades: Short range, extremely high damage. Seriously, fucking huge damage. They're not that good for much except sitting on top of JPs to ambush attackers, though, since even with extremely high-end stealth systems you'll still be detected before you can get close enough to alpha strike anything. Damage profile is big craters.
But the overwhelming rule is that for offensive purposes missiles are almost always objectively better, due in part to the unfortunately nature of combat in Aurora.