Personal Input:
TEC:
Strengths: The Trader Emergency Coalition was formed of...well, traders - as such, their econ is pretty damn powerful. Repair bays can be just phenomenal once fully upgraded, and the mighty Novalith allows you to nuke planets from across the galaxy - although it might take a while for the warheads to get there. The Akkan's reworked colonize, as Darvin noted earlier, means they have a reason to get the Akkan right off the bat.
One other thing to note about the Novalith is that in Entrenchment, it lowers the population cap at a planet for a good 10 minutes or so, meaning that if an enemy decides to re-colonize the planet again, they won't get as much income as they should. A "Novalith Network" is a good way to quickly make income drop to almost nothing - I like to call it "The Fan of Death," as an aside. Their Ogrovs are also the ultimate Starbase-Killers (which is all I use 'em for, mainly) - 10 or so pointed at a starbase allows them to massacre anything within seconds.
Weaknesses: It's been noted multiple times already, but while they might not SUCK in any categories, they don't really EXCEL in anything either - jack of all trades, master of none. After getting a better look at Argonevs, I'm inclined to agree that they probably have the worst of the lot - on the plus side, most of the stuff you research for it can be maxed out with the 8 points, whereas you'd be sacrificing some of that for other things on the Transcencias and Orkies.
What surprises me, though, is that no one's mentioned mines. TEC have to phyiscally BUILD their minefields before they can be put up, and when the enemy's breathing down your neck, it's an inconvenience you REALLY can't afford. It also means they can't put any mines at stars, whereas Advent and Vasari can.
"Tweeners": Insurgency allows you to harrass your enemies regardless of where they are, but they don't care if it's you or someone else - they attack anyone on sight. Foreign Sabatoge causes enemies w/in your culture to build slower, but it's a bit too situational for any practical use unless you're trying to gain a foothold in enemy territory.
Advent:
Strengths: Beware a skilled player with Advent. S/he can run you through with their crazy-high damaging ships and wipe you out before you know what hit you. Illuminators frequently tend to be the culprits. Also, as mentioned earlier, their Caps help each other out with Synergies, meaning if a player micros efficiently, a full fleet can rip an enemy to shreds and keep fighting. The Transcencia is capable of summoning meteors, holding your ships in place, or blasting you to bits with their 20 fighters/bombers. In addition, they can also spread culture and increase allegiance at planets, allowing them to get up to 150%(!) at their homeworlds.
Also, the homing mine drones that Aerias or Hangar defenses can deploy are SERIOUSLY bad news for anyone. They can be deployed anywhere (excluding magnetic clouds or ion storms), and I pity the poor sucker who wanders too close to one of 'em.
Weaknesses: Again, as noted, the Advent probably have the worst economic war machine - tech level 3 just to get access to both metal and crystal research. Illums also require tech level 4 before they become available, and the antimatter dependence of the Advent means that they can be in real trouble if their ships run out. And this goes without saying, but Vasari players can simply MURDER Advent w/phase missiles - again, as noted, Advent tend to have the weakest hull out of the three, and once that drops to zero, kiss your ass goodbye.
The infamous Solanus Adjudicator, often referred to as the "Starfish," also pales in comparison to the Ogrov - to deal the most damage, you need to have 5 targets in range, which is pretty much a suicide attack unless you're attacking a VERY sparsely-defended area - and even then, the supply demands are so prohibitive you may as well just build Aerias, outfit 'em w/bombers, and call it good.
"Tweeners": Though the Deliverance engine got bashed, I tend to see it differently. Sure, it might not do any physical damage, but a massive "Deliverance Network" can keep the culture from receeding quickly. Coupled with Eyes of the Converted, and suddenly pairing this with Clairvoyance seems much more attractive. Also, in Entrenchment, your ships' attack power is boosted in a well hit by a DE. It is, however, only really feasible if you've got a coordinated effort.
Vasari:
Strengths: Phase missiles. Phase missiles. PHASE MISSILES. These weapons give the Assailant, fighters AND bombers, and all caps that use them a chance to IGNORE shields. Against the Advent, that's a very bad thing...for them. The Orkulus, quite simply, rocks the damn house. Mobile, 20% increased damage, and by and far the strongest hull/shields of the three, if you allow a Vasari player to build an Orky in your grav well, you deserve it, quite frankly.
The Orky's mobility has led me to another insight - when you're certain of where the enemy's coming from, you just re-position the SB and call it good, something you can't do with the other two. Phase Gates allow you to get instant reinforcements on call at any time (not to mention the ability to get to hot spots quickly), and they can get an additonal 10% increase to their fleet size. The Sivuskras Ruiner, the Vasari's mine-laying ship, allows deployment of mines ANYWHERE except magnetic clouds - if you find choke points, then, by God, USE THEM! Gravity mines keep your prey from escaping for a little bit (side note - they HOME), and the explosive mines are just fun for all ages.
Weaknesses: Once again, as noted by others, Vasari ships are the most expensive of the three - both in terms of materials and supply demands - their basic light frigate requires SEVEN supply, as opposed to the TEC's five and the Advent's four (or is it 2? I can't remember). Their heavy cruisers, the Skarovas Enforcers, also are in desperate need of some well-deserved attention, as they often find themselves outgunned in spite of their increased cost.
In spite of all the good things about the Orkulus, there are, of course, problems with it, as well. For one thing, it's the Vasari's only viable anti-structure cruiser, which wouldn't be a problem...at least, if not for the fact you have to build it - in an enemy GW, no less - first. Also, I've heard reports that the +20% damage app doesn't work properly, unless IC fixed it already, but I won't stir that hornet's nest for now.
"Tweeners": The Orky may be disadvantaged that it can't attack fighters...but unlike the Argonev or Transcencia, it can chase down the carriers sending said fighters. If 4 Orkies are at a star, try to spread them out so they can cover each other if carriers become a problem. The Kostura Cannon is kinda "meh"-ish in terms of function (structures disabled for 3 mins, ships for 15 seconds) - but by the time of Entrenchment, Vasari scientists have managed to equip the cannon's warheads with Phase Stabilization Generators, creating a temporary Phase Stabilization Node for you to jump to, meaning that if you've got a small foothold established, you can instill terror in your enemies at any moment of your choosing.
That should cover it.
