Why do some fighter jets have movable horizontal stabilizer instead of elevators installed on the stabilizers?
All parts of an airliner's horizontal surface move, not just the rear part. The rear part, called an elevator, can move much faster and is for maneuvering. The forward part, called a (trimmable) stabilizer, is for trimming and moves slowly. It is moved in response to changes in loading, speed or flap settings and positions the tail surface such that only small continuous elevator deflections are needed. It doesn't need to move fast - high load factors would upset the passengers and overload the structure.



A330 port tailplane root (picture source). Note the markings which show the range of incidence angles covered by the trimmable stabilizer.

Benefits of a stabilizer-elevator configuration:


Camber: The elevator deflection changes the camber of the airfoil of the tail surface and makes the production of the intended lift change more efficient. If the elevator deflection is supposed to create a downforce, negative camber is produced and vice versa. This reduces the drag which is created in order to maneuver the aircraft.
Lower control forces: By moving a smaller surface, less hydraulics power, and on small aircraft, less muscle power is needed for the same moment change than with a full-flying tail or all-moving surface. Makes sense, right?
Better tailoring of the hinge moment derivatives: Two effects are important for getting the control forces right: The change in hinge moment over deflection angle ($c_{r\eta}$), and the change in hinge moment over angle of attack ($c_{r\alpha}$). With tabs, the right elevator hinge position, nose shape and control horns, both can be tailored individually, while a full-flying tail or all-moving surface will give the designer less freedom to manipulate both independently. Note that full-flying tails only emerged with hydraulic controls, because only those systems can manage the control forces of all-moving control surfaces at high speed.




Horn balance and overhang balance on control surfaces (picture source)

Benefits of a full-flying tail surface


By avoiding a contour break due to a flap deflection, all-moving control surfaces can avoid the shocks which would otherwise occur at high subsonic speed. This is their main benefit for combat aircraft.
At supersonic speed, the uncambered surface will produce less drag, so for supersonic speed the all-moving control surface is more efficient.
By moving all of the tail surface at high speed, an all-moving control surface will produce the highest rate of moment change over time possible. To produce also the highest sustained moment change, however, it needs to be larger than a comparable stabilizer-elevator combination because it forfeits the benefits of variable camber.
The all-moving control surface has no gaps which could add radar reflections, allowing for a more stealthy design.
Lower mechanical complexity. This is somewhat offset by the need for much beefier hydraulics.