I haven't read all of the replies, so bear that in mind when you read this. Also, I will be getting more at the advantages/disadvantages that refer to online play.
TEC
Greatest Advantage:
The direct nature of the capital ship abilities, and more specifically, The Akkan's colonize ability. This ability has renewed my faith in TEC. I know there's a lot of things about tec that people could say is the greatest, but in terms of indirect impact in the early game, the Akkan to me does it all. The free mine and income bonus is actually MORE than the Progen econ bonus brings. I'm serious. Crunch the numbers.
Greatest Disadvantage:
I have to go with both starbases and pji's being at tier 4. I know people are probably saying weak frigs, but they're cheap and take up 4 ships slots. C'mon. That's strength in numbers.
What I'm getting at is if you want to be defensive or set defensive traps, you're kinda at a disadvantage. It takes 4 labs for a PJI or a starbase, two buildings the other races have easier access to.
Advent
Greatest Advantage:
Bugged Illuminator. Next question.
Greatest Disadvantage:
Lackluster economy. Going econ with Advent isn't impossible, but Vasari and TEC have better techs/researches that outclass the Advent
Vasari
Greatest Advantage:
I have to say I'm split between phase missiles and a rejuvenated capital ship core. The phase missiles make it so no capital ship is safe, and the new cap ships have been upgrades so much that they actually compare to tec/advent.
Greatest Disadvantage:
Weak units all around. Weak lrf. Weak Scout. Weak light frig. They rely on trying to get by with more resources from neutrals to balance out the fleet by upgrading it and making it bigger earlier, when in reality, their units....SUCK.