You are correct when you say Vasari are slow at first. Stats per fleet supply point on most of their ships are worse than most of their TEC and Advent counterparts, so the Vasari can't use brute force very well. What really sets them apart is a vast array of really awesome research and special abilities. Phase Missiles are easily the best weapon in the game once you start upgrading them, and can negate the inherit firepower disadvantage of ships that use them. Phase stabilizers are amazing as it allows you to defend all of your planets with a single defensive fleet, which give the Vasari a huge advantage on large maps. Many pro strategies rely on Subverters to disable large portions of the enemy fleet, which is devastating enough on its own. I could go on for a very long time.
However, almost all of those things requires research, which is most certainly not cheap. To compensate, the Vasari have two early game advantage you have to exploit to survive. The first is the quality of their capital ships. Their carrier, battleship and colony caps are all AAA caps, and they can handle a variety of situations. The other is that their scout can capture neutral mineral extractors (other factions need a colony frigate), which gives them an early boost in resources, but this has been diminished after quick start was introduced. Still, currently among good players many thing the Vasari is the best overall faction.
In short, the TEC is almost the opposite of the Vasari, as their units have better stats but not many unique abilities. They rely more on out producing the enemy than the unit strategies the Advent and Vasari need to deploy.
The Advent are more unique. In general their strength relies on other ship types buffing the fleet as a whole to create "synergies" that are far more than the sum of its parts. Another way of looking at it is that most Vasari abilities make their opponent weaker, while the Advent prefer to empower their own units (though each has a couple of abilities of the other category).
