When you're not battling, your Digimon will not go back into balls, but can head to your digi-farm where, if you've supplied them with the right equipment, they can hit the gym and train up on their own. We've never seen Pikachu on a Soloflex, have we?
Finally, the two versions of the game will feature the same set-up story, but things diverge from there. It starts with an earthquake and a virus that causes many digimon to devolve into more primitive forms. In Dawn, you're part of the light side (lots of holy and animal digimon) and you set off to battle the dark side, whom you think caused the catastrophe. In Dusk, you're already on the dark side (favoring machine and bug types) and take a different path through the game because you're fighting the light side, whom you blame for the attack.