I play only Vasari now, after I realized that they were the best race by far for the type of game I like to play (10-player FFA against Hard AIs on a 115-planet map). They're definitely NOT underpowered, by any means. Here are the high points, as I see them:
1> DEFENSIVE MOBILITY. Phase stabilizers allow the Vasari player to quickly shift his fleets to where they're needed, making them far more able to react to attacking enemies. On a large map, it's nearly impossible for Terran or Advent players to keep enough ships around each threatened world to make a difference, but Vasari don't have to bother with that, and can concentrate all of their firepower into a handful of "offensive" fleets that can be pulled back quickly if needed. (I generally make four 4-cap fleets and use two as roving defenders while the other two hit my enemies where they're weak, and then rotate which two I use on a regular basis.)
The flip side of this one is that the Vasari need to shift fleets for these purposes, more than the other races, because their static defenses simply can't hold off an attacking fleet for nearly as long as the others'. Advent can depend on the synergy of their defenses and the +10 tac points they can research, TEC defenses have more raw firepower on each turret or hangar, but Vasari defenses will be weaker even before subtracting the 10 points for each stabilizer.
2> OFFENSIVE MOBILITY. The ability to use a Kostura for temporary phase lanes is part of this. The ability to use a Marauder to create a node is another. But there's more than that; if you send a fleet to bypass the enemy's outside systems and then capture the next system in line, you can create a phase stabilizer in THAT system to allow all of your other fleets to bypass the crustal defenses. (I often do this when I capture a pirate system deep within an ally's territory; if the ally turns on me, I have a staging base inside his defenses.) Or consider that Vasari have a research tech that reduces the jump radius of gravity wells, allowing them to jump in closer OR jump out from further in... this noticeably reduces the time needed to move across large maps, and makes it harder for enemies to engage fleets that are just passing through to hit a weaker planet.
3> FIREPOWER. Vasari have research that gives them up to 15% more fleet points for the same upkeep, allowing them to field substantially larger support fleets. Their superweapon can have a tremendous impact on combats, their techs are more offense-based, their culture increases damage, all that sort of thing. Phase missiles are also the best weapon system in the game, IMO, as shield mitigation becomes a more pronounced issue as fleets get larger; every other weapon type is basically interchangeable, but Vasari get the TWO weapons that have distinctly different mechanisms (with the starbase Disintegration Beams being the other).
4> STARBASES. The octopus is just great. The weapons mix is good, the defensive specials are handy, and of course, it's mobile. This means you can build it at the edge of an enemy's gravity well instead of trying to sneak a constructor deep in, and you can force it to move into the center of an attacking fleet so that it can use the banks on all sides at once for massively increased damage.
5> CAPSHIPS. Besides just looking cooler, Vasari capitals have a few noticeable advantages over other races. They ALWAYS fire forward as primary, which means no time is wasted rotating for a broadside. There's no assymetry, where one side has more guns than the other, and most of their weapons will be in the forward mounts (especially on the desolator). They also have some of the best activated capship abilities in the game, nearly all of which are fine to set on autocast. For instance:
> Having an AoE repair cloud (as opposed to the various single-target repair abilities in the game) is HUGE when facing starbases, as you'll have a dozen or more ships getting damaged at one time; adding a couple hundred HP to each of a dozen Enforcers can mean the difference between winning without any destroyed ships, or limping out without half your fleet. It's also good for keeping your bombers and fighters alive, since their HP are just high enough to take a few hits before dying (unlike the Advent "Wet Kleenex" fighters).
Many of the other capship abilities are useful AoEs like this; the egg's ability to prevent groups of enemies from jumping has let me kill quite a few enemy capships that otherwise would have escaped.
> Nano Disassembler and such are great against single, heavily-armored targets like Starbases, and so a mixed group of Vasari capships will have great synergy. I tend to make four fleets, each of which has an egg, a devastator, a desolator, and a carrier, and even fully-upgraded starbases won't last long against their mix of abilities (especially Assault Specialization).
> Most of the level 6 ultimates are excellent abilities. Not as damaging as the Marza's barrage, for instance, but they're all still extremely useful. In fact, the only Vasari capship abilities of any kind that I don't use on a regular basis are the Marauders' abilities. When playing the other races, I'm always struck by how many of their ships' abilities are situational or don't have a clear impact.
6> THE LITTLE STUFF. Scouts that can capture neutral extractors, colony frigates that can build starbases, fighters and bombers that can take a beating without being destroyed (especially good with the new 50% penalty to build rates), single techs that boost both metal and crystal harvesting instead of needing one tree for each... it all adds up.
Bottom line, as others have said: for the other races, it's often that the capships are built to support a well-balanced fleet of frigates and cruisers, while with the Vasari it's the other way around. You build the capships, and then attach some cruisers or frigates to support them while the capitals do the main killing. It takes a little getting used to, that's all.