When an LTI system is NOT completely controllable/observable, does the linear differential equation governing the system give the internal or the external description of a system?
External. That's the whole point: The behavior of any LTI system can be completely characterized by it's transfer function, impulse response, differential equation, etc. (which can all be converted into each other).
That's a "black box" description and gives no information on the internal details of the system. There many different physical realizations of the same system, but you can't tell from just the outside behavior