Although I still haven't seen the movie, reviews would seem to indicate that the "new torpedo tubes" were pure obfuscation - they were part of the Marcus scheme to blackmail Khan into continuing to serve Starfleet interests. Their only actual function was to provide an excuse for Marcus to smuggle the 72 helpless Augments aboard the ship and then threaten to kill them unless Khan cooperated.
The "old" tube in the neck probably always provided the usual range of services, including all the sublight and warp launches needed. The "new tubes" might have been sublight-only deployment chutes from the sound of it, superficially plausible for use in planetary precision bombardment (the role Marcus claimed they would serve in the Khan hunt), so Kirk would buy the lie easily enough.
I prefer to think that Enterprise carries at least two different types of torpedoes on board, necessitating at least two different types of tubes. The side-mounted ones are probably used -- under normal circumstances -- to launch drones, mines, or short-range point defense interceptors.