What are the reasons to evolve from trim tabs to moving the entire surface (e.g. stabilizer)?
Trim is used to zero the stick forces at the desired flight condition, and with the small, slow early airplanes, an adjustable spring somewhere along the control cables would do. When speed increased, trim tabs were introduced, because their trim force goes up and down with dynamic pressure, whereas the spring would always create the same force. This would require re-adjustment with every speed change.

With the increasing size and speed of airplanes, manual control required ever more complicated means of force reduction (Flettner tabs, spring tabs), and the relative chord length of all moving surfaces shrank to allow for more control with less force. The moment around the hinge of a control surface increases with the square of the chord, but the lift change for a given deflection angle increases only with the square root of the chord.

In parallel, wing loadings increased, which required high-lift devices to keep take-off and landing speeds down. These add a lot of lift in the rear part of the wing, creating a strong nose-down moment, and now a simple control deflection would not suffice to trim the airplane. The reduced chord length meant that high deflection angles would have been required for trim, reaching far into the region of deflection angles with declining effectiveness. That is when adjustable stabilizers came up: The trim change with all configurations was simply too big to be covered by the elevator alone.

Transsonic flight was next, and here the adjustable tail incidence is crucial to trim the aircraft trough the transition from subsonic to supersonic flight. A contour break from a deflected elevator could cause shocks which could produce elevator reversal. The dynamic pressures of jet flight put an end to manual control - the last, famous examples were the Canberra or the Dash-8 (Boeing 707 prototype). Now hydraulic actuators were used, but still it makes sense to use aerodynamic means to reduce control forces, because the size and power requirements of the actuators can be reduced with clever hinge moment reduction. The trim tabs were initially moved by cables, but soon they used electric actuators.

Since rudder and ailerons are not affected when the high-lift devices (Fowler flaps, slotted flaps) are deployed, they still use simple trim tabs. Newest developments are electric control surface actuators which can help to save mass and avoid the messy hydraulics. The RQ-4 uses electric actuators because the hydraulic fluid would have to be pumped continuously through the lines in order to keep it warm enough at the high operating altitude. Only the landing gear still uses hydraulics for retraction.