What are the advantages of a Trimmable Horizontal Stabilizer?
The main advantage is smaller elevator deflection angles. This comes handy in two cases:


When high-lift devices are deployed, the center of pressure on the wing shifts backwards by up to a third of wing chord. Fowler flaps add wing area aft of the trailing edge, and slotted flaps are able to generate high suction peaks. The result is a massive change in trim, and the empennage now has to generate generous downforce. Changing the lift on the empennage by elevator deflection alone will exceed the maximum practicable deflection angle and leave no margin for control. By adjusting the stabilizer incidence, the elevator can be held near its neutral position and has reserves for control.
In transsonic flight the elevator might not always have a linear characteristic. The contour break due to an elevator deflection induces shocks which in turn lead to flow separation which reduces the control effectivity and can even reverse the control characteristic. Since the transition from subsonic to supersonic flight shifts the center of lift backwards, the empennage needs to add downforce when the aircraft accelerates in the transsonic speed range. An elevator deflection might not be able to produce the desired lift change, and only adjusting the stabilizer such that the elevator can be held neutral can restore trim and control effectiveness.


Older airliners from the propeller age had lower wing loadings and less powerful flaps. The center of lift on the wing changed less with flaps, so a fixed stabilizer was sufficient. But once the wing loading goes up to jet levels, and the wing is fitted with slotted fowler flaps, a moveable stabilizer is unavoidable.