When you're more experienced and feel good about the mechanics of the game and understand the three races better, be sure to come play the game in online multiplayer--what some people call "The Real Sins Game"--(click the Ironclad Online buttom) and I'm sure someone will demonstrate to you the power of the Unity! (Really, once you get tired of playing against the predictable AI, which online players don't even count as a player since it's so awful compared to human players, you should come play online. You'll get to enjoy more intense, competitive, and suspenseful games as well as having human teammates and camraderie and you might even make some friends.)
The Advent have a couple things going for them. Overall, they do have the best capital ships though you probably won't want to build more than one or two for the online multiplayer game since people tend to focus fire on them and destroy them. The two capital ships are the Progenitor Mothership which can repair the shields of an unlimitted number of your ships as well as give you the very valuable 20/40/60 colonization bonus and the Halcyon Carrier which can increase the amount of damage your ships do (up to 22% I think) and it also has a good anti-strikecraft capability (aka "the Pimp Slap").
The Advent also have the best and most powerful LRM (long range missile) ship in the game called the Illuminator. In fact, people often complain about a tactic called Illuminator Spam. Illuminators are more resiliant than the TEC LRMs and the Vasari Assailants and can fire in three directions at one time, so you try to move a pile of them into the middle of enemy forces and structures and let the rear and side beams do some of the work. This is a significant early-game advantage for the Advent, especially when fleets number under 20 ships. Combine the Illuminators with the Progenitor's shield restore and they'll beat up on other fleets in the early game (which is very important since many online multiplayer games are decided early-on). Their heavy cruiser, the Destra, is also good.
One other overlooked advantage is that they probabl have the strongest and most resiliant scout ships in the game and they only use 2 fleet supply points (Vasari's use 3). Thanks to Raging Amish, an increasingly popular online multiplayer tactic is for people to spam these out in the early game (only cost 200 credits) and use them to counter enemy LRMs. You don't need a single military lab to do this.
They also have the strongest culture in the game, which, while easy to overlook, can be rather important. They have some other research-based advantages as well in addition to the disadvantages that every race has. In fact, a great many skilled and experienced players choose to play Advent.
As I see it, Advent's main disadvantages are:
They need 3 military labs early-on to make their Illuminator LRM ships as well as to put up starbases (in Entrenchment). In order to both produce the LRM ship and to be able to colonize ice and volcanic (2 civic labs for each race), Advent needs 5 labs. TEC needs 4 and Vasari needs just 3. (Note that TEC needs 5 for starbases and Vasari just 4 for starbases.)
They cannot improve the amount of income they receive from trade (unlike TEC and Vasari).
They have weak hulls which makes it easier for the Vasari to take them down with phase missiles.
The first capital ship that you would build for them (always the Progenitor) is the worst of the three first capital ships for all of the races in the areas of planet bombing and one-on-one capital ship-to-capital ship combat. (If you play TEC, your first capital ship should be the Marza for its Level 6 Missile Barrage ability, and if you play Vasari your first capital ship should be the Evacuator for its Colonization, Nanites, Phase Jump Inhibition, and Level 6 Planet Suck abilities).