The Ancient Geek
In an airliner with a THS there is no separate elevator or trim tab, the entire horizontal stabiliser is moved by jacks.
In the Airbus the side stick is effectively a fine control which moves the THS within a limited range. The trim wheel is effectively the coarse control with full authority from full up to full down. If sidestick inputs reach the limit of their effective range the autotrim system system acts, think of this as moving the small window of movement available to the sidestick up and down within the full THS range.
That is more than oversimplified, it is just incorrect. Each airbus has two full elevators attached to the THS, just a standard layout, no trim tab though.
The SS does neither move the elevators nor the THS, but it orders a change in flightpath to the computers, which deflect the elevators in order to get the flightpath changed as desired. The autotrim trims the THS into a position, where the load of the elevators is removed.
The Ancient Geek
If a pilot holds the sidestick in the full up or full down position the autotrim will obey and drive the THS to its limit. Note that if the pilot now releases the sidestick to its neutral position the sidestick now has a small window of effectiveness at the limit of THS travel.
Again, itīs the computers who use the elevators and the trim. If the SS is returned to neutral, no further flightpath change is ordered, 1 g is maintained and the computers position the elevators and the trim to achieve that. In the end the elevators are neutral and the THS has found its new trim position.
In case of AF447 Law Alt2b was active without protections. Therefore in a decreasing speed situation the computers would have ordered a NU command to the elevators and finally also to the THS in order to maintain 1 g flightpath even without the command from the SS.