I'm confused as how much my starting class matters. Seems like all your power come from your BOs right? Is the skill tree different? What does my engineering character gain from being a engineer?
On the ground, the difference is in the kits you can equip to your character. Kits essentially let you use the bridge officer skills, but you can swap them in and out as you collect new kits, rather than invest skill points into each skill. Tactical kits let you use grenades or improve your weapon damage/accuracy, things like that. Engineering kits let you fabricate turrets, cause enemy weapons malfunctions, etc. Science kits have a lot of healing abilities, plus a number of different types of attacks associated with them. Naturally, you can only use a kit that corresponds with your character class.
Each ship tier tends to be better suited to a character class as well, though it is possible to mix and match. Escorts are best for tactical officers, Cruisers for engineers, and Science vessels for science officers.
Finally, as you rank up, your character will gain access to various skills related to their class. Tactical officers learn Attack Pattern Alpha (buffs various combat stats for a time), Engineers can generate a Cover Shield, etc.
Some Bridge Officer powers can only be gained if a captain player character 'trains' a Bridge Officer in them. If you look at the captain skills table, many of the skills say they unlock the ability to train Bridge Officers in a given power.
For instance, the only way you can get Extend Shields III or Engineering Team III on a Bridge Officer...is to be an Engineer captain. An Engineer can give his Engineering Bridge Officers those powers, if he maxes out the relevant skills.
That said, all of the player-trained BO powers like these are Lt. Commander and Commander grade. This means only an Engineer in a Cruiser gets the benefit. An Engineer in an Escort or Science ship can still train BOs, but the BOs can't actually use the trained powers because Escorts and Science ships only have Lieutenant Eng BO slots.