The faster an aircraft is, the wider range of elevator deflections it needs. And so do aircraft with large change of weight (amount of fuel used) during flight. However larger elevator deflection causes higher drag. It is more efficient to move the whole horizontal stabilizer instead. Therefore jet aircraft generally have the forward part of the horizontal stabilizer movable.
Now it would be possible to simply have the whole stabilizer move. And supersonic aircraft generally have just that. It is called stabilators. However having the stabilizer split lends itself to elegant solution for trim.
This takes advantage of the fact, that the aerodynamic forces tend to put the elevator in the position of lowest drag almost straight (slightly bent in direction of less lift) behind the forward part. So when the forward part is moved, the lift changes without changing the force on the elevator. So the elevator is used for the small adjustments where the elevator is intended to return to the neutral position, which the aerodynamic forces help with, and the stabilizer is used to trim for current airspeed.
In aircraft with mechanical controls, the elevator force (or a fraction of it via power drive) acts directly back on the control column and the pilot can feel it, while the horizontal stabilizer is actuated via mechanism that holds selected position (may be electrical with jack-screw like on DC-9 or hydraulic, but with no feedback) and connected to the trim, which is usually pair of large wheels on the sides of the centre console.
Now in Airbus A320 and all newer models the side-stick does not have mechanical link and the pilot does not feel the force on the elevator anyway. In normal law, the ELACs automaically adjust the elevator and stabilizer so that with side-stick in neutral position, the aircraft maintains 1 G vertical acceleration and therefore straight flight (at any flight path angle). Since the trim wheel does have direct mechanical link, it turns as the ELACs (or SECs) adjust trim.
However the distinction between elevator and trim is still useful in case of failure. When the system degrades into direct law, the side-stick position corresponds to the elevator deflection and the trim wheel position corresponds to the stabilizer position. Since the side-stick still returns to neutral position, the distinction of using the elevator for momentary adjustments and trim to balance the aircraft to fly (approximately) straight at current speed holds (just because the side-stick is spring-loaded, the force needed to deflect it does not increase with speed as it does for mechanical controls).