I've been thinking about the directed energy weapons in Sins of a Solar Empire: how do they function, how do they relate to one another, and what do they signify in terms of the technological development of the three factions?

I'm aware of the pitfalls in thinking too much about this issue. Allow me to forestall a few arguments with the following disclaimer:

1. Sins takes place in a universe in which psychic powers and deflector shields exist, faster-than-light travel is possible (throwing causality RIGHT out the window), and phase space is more than a representation of potential system states. Suspend your disbelief.
• I'm going to assume that deflector shields evolved from the need to shield ships from space debris during relativistic flights. Remember: shield mitigation is explained in the manual as a function of the shield changing frequencies and adapting to incoming weapons fire.
2. I'm aware that plasma is an ionized gas, and a "plasma weapon" would probably look something like a flamethrower. Consequently, I'm going to assume that plasma weapons in the Sins universe use some sort of containment field (gravitational, electromagnetic, or phase-space) to contain the ionized gas in either a projectile, beam, or wave form.
• I will note here that both Advent beam units (Illuminators, defense platforms, bombers, and capital ships) and certain TEC units (the Kol Battleship and the Argonev starbase) appear to take advantage of the same weapon system: the weapons have a similar appearance (a thick beam of coherent light, though the Advent's are blue where the TEC's are orange) and are referred to as "Beam" type weapons in the the tooltips and research trees. I'm going to assume that Beam weapons are, in fact, plasma beams.
• It has been brought to my attention that beam weapons may actually be particle or ion beams (a-la Homeworld). I assumed that since pulse guns are pulsed plasma guns that pulse beams would be pulsed plasma beams. That said, it is possible that the pulse gun/pulse beam research tree is connected primarily because they both focus on the application of pulse technology to weapons systems beyond lasers, namely plasma guns and beam weapons.
  
  Suffice to say that, regardless if beam cannons are ion beams or plasma beams, the Vasari are a step ahead of everybody in both the plasma burst gun and beam weapon departments, since they use pulsed versions of both of those.
3. I'm going to assume that "pulse lasers" are basically rapidly modulated lasers, represented by a pew-pew projectile graphic in the game to keep things visually lively, but otherwise a straight-forward laser beam that changes frequencies and intensities algorithmically to counteract shield mitigation.
• I imagine that shield mitigation against a conventional, non-pulsed laser weapon would approach 100%.
4. I'm going to assume that the Vasari have reached the limits of conventional technology (unassisted by psychic abilities); I will likewise assume that both the Phase Jump Inhibitors (which the Vasari can replicate with nanotechnology but cannot understand) and whatever drove them from their home systems represent UNCONVENTIONAL technology. 
5. If you examine the opening cinematic, you will note that whatever drove the Vasari from their home systems emerged from a Viturska Imperial Lab dedicated to "Experimental Transport" research.

Nevertheless, I'm willing to speculate.  The following constitute my analysis of the Sins of a Solar Empire energy weapon hierarchy.

In the Sins universe, the three factions appear to be roughly balanced:

• The TEC combines armor that is extremely durable for its cost with mass production to compensate for their lack of high-tech weapons. Regardless of TEC stats in-game, on a pound-for-pound basis it's doubtful that TEC armor can match Vasari armor, which is nano-engineered, self-repairing, incredibly light (high-tier Vasari armor actually reduces ship mass) and versatile. Fortunately, the TEC don't HAVE to match Vasari armor on a pound-for-pound basis: they have the luxury of producing SLIGHTLY inferior, conventionally engineered armor, at MUCH higher rates of production. TEC armor probably represents a compromise: the best protection they can manage for a reasonable cost, cranked out in such large quantities that the TEC can field two well-protected ships for every excellently protected Vasari vessel.  Aside from autocannons, missiles, and pulse lasers, the TEC are beginning to break through into sustained beam technology, as evidenced by the beam cannons on the Argonev starbase, the Ankylon Titan, and the front of the Kol Battleship. These weapons are largely primitive compared to the Advent: the Kol Battleship's beam weapons are mounted in a spinal configuration and must be arranged in a cluster to provide adequate firepower. That said, they represent a step toward more robust technology in TEC fighting ships.
• The Advent have used psitech to further refine laser weapons, providing the Advent with pulse lasers that are more than 60% more efficient than their TEC counterparts (established by comparing the performance of the single pulse laser on the Disciple Vessel with the pair of pulse lasers on the Cobalt Light Frigate).  The Advent have also made significant advances over the TEC in the field of sustained beam weaponry. Unlike the TEC, the Advent are not restricted to mounting beams on star bases or in spinal arrays. The Advent have miniaturized beam technology to the point that beams can be mounted in a defense platform, frigate hull, or even as a broadside weapon, as demonstrated by the Halcyon Carrier.  Further advancements yielded short-barreled plasma weapons that may be fired in bursts, widely utilized in capital ship turrets and as the primary armament of the Destra Crusader heavy cruiser. These burst-firing plasma weapons represent the pinnacle of Advent plasma weapons research.
• The Vasari, through either trial-and-error, engineering genius, outright technological theft, or some combination of the above, have long since mastered plasma technology, to the point that plasma weapons constitute basic armament on the vast majority of Vasari ships.
• The most common plasma weapon in the Vasari arsenal is the pulsed plasma gun, being present on no less than five out of ten of their most common frigates and cruisers. By applying the basic principles of pulsed weaponry to burst-firing plasma weapons, a single frigate-sized pulsed plasma gun can achieve nearly 63% of the destructive power of FOUR cruiser-sized burst-firing plasma weapons. The widespread usage of this weapon is limited by the fact that pulsed plasma bursts do not function well at large scales. To achieve appropriate destructive levels, pulsed plasma guns must be mounted in clusters, as demonstrated by the Rankulas battleship.
• To compensate for the lack of scalability inherent in pulsed plasma guns, the Vasari applied the principles of pulsed weaponry to beam technology, as well, yielding pulsed beam weapons capable of delivering their destructive payload in a fraction of the time compared to sustained beam weapons. The average pulsed beam is anywhere from 12% to 100% more destructive than a comparably sized sustained beam, as demonstrated by analyses of the damage inflicted by the three flank-mounted pulsed beams on the Kortul Devastator and the four similarly sized broadside sustained beams mounted on the Halcyon Carrier.
• For weapon emplacements that are too large for pulsed plasma guns and too small for pulsed beams, the Vasari rely on a much older weapon: the plasma wave cannon. Prior to the development of pulsed plasma technology, plasma wave cannons were the predominant directed energy weapon on all Vasari ships.  Plasma waves are modified plasma bursts engineered to distribute plasma over a much broader, thinner area, slicing through multiple decks rather than punching small holes in the target vessel. In practice, plasma wave cannons are approximately 21% more destructive than similarly sized plasma burst cannons.
• Sustained beam cannons have been all but phased out of the Vasari navy.  The exception is orbital bombardment, wherein they channel a large-scale sustained beam towards a planet's surface. The primary advantage here is the negation of the need for ammunition and supply lines: any Vasari ship capable of orbital bombardment can do so as long as the ship is functional (as opposed to the TEC and the Advent, who use nuclear weapons and kinetic bolts for orbital bombardment, respectively).  I like to think that the Vasari are the only race in the galaxy who have managed to field a directed energy weapon capable of penetrating a planet's magnetosphere (which deflects a truckload of radiation all the time).
• The Vasari disintegrator weapon is technically not an energy weapon at all. Rather, it is an application of the same nano-assembler used to construct their ships. The disintegrator works by spraying the target with a cloud of nanites which flay the target layer by layer, deconstructing the target at a molecular level. Though short ranged (owing to slow speed of the nanite cloud and the difficulty in maintaining cohesion), it is a devastating weapon.

Surprisingly, that only took me approximately an hour and a half to write. It was worth every minute.

TLDR:

• TEC: [Pulse lasers] + [primitive sustained beams]
• Advent: [Psitech-enhanced pulse lasers] + [advanced sustained beams] + [burst-firing plasma weapons]
• Vasari: [Pulsed burst-firing plasma weapons] + [pulsed beams] + [burst-firing plasma weapons modified to fire wave-shaped plasma bursts]

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