The story is unfinished and ends soon after you make the choice. It has no effect on anything right now (chat UI color changes but that's about it), but if you choose one of the generals, you will be able to call them to your party. Suong should also be in her powered up state if you chose her. Joining White does nothing right now. He will still let you get ichor powers and sell you items even if you didn't join him.
All of this will change in the future however and there will be pretty big consequences for joining any of the four characters.