Many aircraft - particularly large commercial airliners - have both an all-moving tailplane (horizontal stabiliser) and elevators. (They also have a small third moving element on the trailing edge of the elevators too.)
As far as I can tell, all these surfaces do exactly the same thing, ie cause a change a pitch. Indeed this answer confirms that both stabiliser and elevators have the same effect, although hints there may be differences at transsonic speeds, something that doesn't apply to most airliners.
I understand trim is concern, but ultimately you're just trying to point the nose up or down. Especially with computers calculating optimum angles, having two control surfaces, which may counteract each other, seems to add a lot of complexity.
Of course there must be some advantage to having both, I'd like to know what that is?
737 empennage (source)