I think one of the reasons why attitude = speed and power=rate of climb is thought to new students, is because of the inerita of most aircraft.
The concern being that an inexperienced pilot finds themselves veryslow on final, and attempt to correct the situation simply by adding a bit of power. Even adding lots of power will be of no use in most aircraft, unless you fix the attitude of the aircraft.
If you get attitude=speed right in your head, then your first reaction will be to push the nose down which will be the quickest way of increasing your speed. (If you are experienced you'll also fix the power at the same time in anticipation of changes needed as a result of the pitch change.)
If you learn power=speed, then your first reaction might just be to increase power and it could end in disaster.
As you get more experienced you'll find yourself able to use both and thinking about both all the time (and eventually as PilorDAR says, eventually not thinking about either, but doing both by instinct).
Your instructor doesn't want you to do one item, then wait before doing the other. They want you to do both at the same time, but want you to get your instinct to be "if slow push the nose down" as it's the action that is likely to result in the fastest acceleration. Increasing power alone might be ineffectual if your nose is already too high.