You gain a spawn point, they lose a spawn point. Kinda like a checkpoint I guess. Also it's a place where you change your class.
We also know that there are dedicated health and ammo Command Posts. I assume that means if there isn't a Medic or Soldier nearby and your team has that Command Post, you can go there to heal up/resupply.
Regardless, being able to change your class in the field is vital to your team's success, depending on the objectives and level strategy; having a forward Command Post is absolutely never a bad thing.
Operatives are the only ones that can steal Command Posts, and can have an Ability that can fortify Command Posts and make it more difficult for enemy Operatives to steal; it will basically lengthen the time needed to capture it and give your team more time to stop the enemy.
Engineers have an Ability called Upgrade that unlocks more features in Command Posts. What those features are, we don't know as yet.