Captain PB is totally correct - it is about energy management.
Checkboard is also correct in that some changes appear to work more quickly than others.
The old "power + attitude = performance" is true and correctly reflects that thrust (power) and angle of attack (attitude) are inextricably linked. But empirically we know that we can place more emphasis on one or the other depending upon our flight path management requirement.
If we are in a fixed power climb or descent, then the only way we can affect our performance, whether it be airspeed or RoC/RoD, is with angle of attack (attitude). Since one of our two key parameters is fixed, we can't totally control our flight path to achieve a specific point in space and we accept that every day - our flight path is not constrained and we happily accept variations in the end point. In this case, we happily control airspeed with attitude (which is really drag with angle of attack) and allow whatever excess power that exists to determine where we end up. And generally, that is the most fuel efficient process.
If we can control both variables but still don't care too much about where we end up, then we tend to select a fixed power anyway.
But the difference from a teaching perspective arises when we wish to constrain the flight path to achieve a particular energy state at a particular point in space. This can arise on climb but is most commonly associated with approaches. The simplest thing to teach someone to achieve a point in space is to use the most effective pointing control, the aircraft attitude - "find your endpoint and fly the aircraft to that point". Clearly, as you push and pull (and, to a much lesser extent, roll and yaw) the changes in attitude (angle of attack) will significantly change the associated drag and hence the speed. As a generalisation, it is more important to be on path than on speed (provided that the speed is not dangerously low) and thrust changes on a constrained flight path have the most immediate effect on speed. Power and attitude remain inextricably linked, but we choose to emphasise them separately in terms of the most obvious outcome - when we fly to a point in space, "attitude controls flight path and thrust controls speed".
I have had enough red wine lubricated discussions with naval aviators to know that they don't share that view - my understanding is that they tend to fix the angle of attack (as directly indicated to them in the cockpit) and therefore their approach speed and use their very responsive engines to fly into the approach path for the wires. And even though I have spent 30 odd years very successfully teaching what I have outlined above, it remains a teaching method rather than some form of scientific absolute.
Having said all of the above, it has always worried me that we have a growing band of aviators who attempt to emulate autopilots in flying the aircraft. Autopilots and autothrottles are simple engineering devices of limited capability and deliberately limited control authority that lack our ability to analyse and predict certain future states. For example, an autothrottle sensing an underspeed cannot predict what you or the autopilot are about to do with the attitude and the resultant drag changes - it cannot decide to leave the power where it is because you happen to be high on path and are about to lower the nose and reduce the drag. Autopilots without autothrottle can only control pitch and roll in response to sensed path error - they can't tell if you are about to stall on the ILS or overspeed the flap. My point here is that they can be good but they can't be (by design) smarter than you. You must know how they operate before you can choose to take any lessons from them.
Just remember, we break things down to teach more effectively - it does not change the physics. When you are really good at flying, the breakdown essentially disappears as you smoothly coordinate all the controls to achieve a smooth flight path with the energy managed to achieve the required targets.
Stay Alive....