I apologize in advance if I am being to basic in this description for your knowledge level. The idea of ''powering up'' or ''powering down'' is a little bit counter-intuitive at first. Even many esperienced sailors do not have a clear picture of this concept. To explain; The amount of force that a sail generates is related to the shape of the sail and its angle of attack to the wind. Angle or attack, or incident angle, is the angle of the sail to the wind. A sail that is fuller (rounder) in shape generates more lift (the term lift is used because a sail is seen as a wing on edge and because, except down wind, the drive that a sail produces pulls the boat from the low pressure side of the sail rather than pushing the sail as one might otherwise assume). The flatter the sail the less lift is generated.
A sail generates lift perpendicular to the surface of the sail with the most lift generated at the luff of the sail and next to no lift generated at the trailing edge. As a result the accumulated forces of a sail can be thought of as having three active components; Drive- which is the component of the force that acts in the forward direction of the boat, Drag which is a component that acts to reduce drive, and Heeling Forces which operate across the boat.
In an ideal sense the goal in sailing is to maximize drive while keeping the other two components under control. For any given boat, in any given conditions, there is an ideal amount of curvature (camber) in the sail and an ideal angle of attack. In light air the goal is to produce the maximum lift that you can regardless of heel angle, etc. But in heavier conditions it is possible to produce too much lift. First of all, except for planning boats, most displacement boats can only use so much drive, i.e. enough drive to push the boat at hullspeed. Second of all with the increased lift comes increased heeling and aerodynamic drag, which may actually slow a boat down and make it less safe and comfortable to sail.
So, the one critical goal in heavier air is to flatten the shape of the sail in order to reduce lift. The other goal in heavier conditions is try to make sure that the sail has the proper incident angle. When you have the sail pulled in too far the sail produces too much heeling for the amount of drive making control and comfort less than ideal.
Which brings us to twist. If you sight up a sail you will notice that if you drew a straight horizontal line
from the mast to the leech of the sail these lines
would not all be parallel. Some would be seen to have at a larger angle to the centerline of the boat than others. This is called twist. When you have a lot of twist in heavy air, part of the sail is over trimmed and part of the sail is often under trimmed and the result is that the part of the sail that is over trimmed is causing the boat to heel excessively.
So talking about the how this applies to actual sail trim underway. In light air you generally can tolerate more power and more twist for a variety of reasons. To power up a sail, the halyard, outhaul, backstay adjuster, and boom vang are eased. The traveler is brought high above the centerline of the boat so that the mainsheet is pulling more horizontally rather than downward. Pulling downward tends to reduce twist and flatten the sail.
As the winds increase, the force on the sail stretches the fabric and in the absence of a boom vang pulls the boom upward, both add curvature to the sail and so provide more lift, increasing drive, drag and heeling. At some point you have too much lift and so you need to flatten the sail out. On a beat you lower the traveler to leeward and tighten the mainsheet. This reduces twist, and fullness in the sail. You also tension the halyards, outhaul, backstay adjuster, and boom vang to further reduce fullness. By the times that the boat is getting overpowered you want to have a very flat sail pointed at a very flat angle to the wind. The traveler should be all the way to leeward and the mainsheet tensioned. The jib
car should be slightly aft of its normal position.
When reaching, without a vang the sail wants to get a lot of twist and to power up in the gusts. This is backward of what you really want to happen. A boom vang, by keeping the boom from rising, reduces twist and so keeps the various portions of the sail at similar angles of attach to each other. This allows you to adjust the sail so that you have just the right angle of attach up and down the sail rather than have one part over trimmed and one part too eased. As a result you have less weather helm and also heel less.
For beginners it seems as if heel equals speed. Generally, heeling does not equate to greater speed. When a boat heels it generally produces greater drag, leeway and is less comfortable to move about on. Beginners usually look at an over-trimmed sail as producing more drive because it produces more heel. In a general sense an over trimmed sail does not produce more drive (or even more lift), just a greater heeling moment due to the greater sideforce of the wind impacting the sail at a deeper angle.
It is only when all of that fails to achieve a comfortable heel and rudder angle that reducing sail area becomes necessary. To keep the terms clear, depowering is reducing the power of the sail area that you have up, and reefing is reducing the area of the sail that you have up. The terms are not interchangeable.
I hope this answers your question.