When you turn by yawing you are skidding the airplane to point the nose to the side, to offset the trust vector to move the plane sideways as it's going forward. The rate of turn you can achieve this way is very low and there is a massive drag of being in a skid as you slither around like a car on ice with the fuselage side presented to the airstream. To the extent that the airplane has a roll/yaw couple (roll caused by yaw) you will have opposite aileron to keep the plane from banking into the skid; more wasted energy serving no purpose.
When you turn by banking you offset the lift vector to move the plane sideways as it moves forward. There is no sideslip so the drag is only increased by the small amount caused by the increased elevator and AOA, and the minor drag of the moderately displaced ailerons and rudder (to the extent they are displaced at all once the rolling action is complete).
Because you are moving through a fluid, a skidding turn is like turning a car by drifting it on a dirt track, a banked turn is like going around on a banked corner, where you don't even have to turn the steering wheel to keep aligned with the road as the car changes direction. Which way is more efficient?
To find out I tried it on my own plane, a homebuilt PL-2. I flew it at 2000 ft, holding 70kt, close to max L/D for this plane, and applied full rudder with a bit of opposite aileron to maintain wings level (not much aileron is required on this plane), adjusting power to hold altitude at 70kt in the skid.
My plane has a manifold pressure gauge even though the prop is fixed pitch, useful for knowing precise power settings. It took 19" MP @ 1900 RPM to hold a skidding turn with full rudder at 70 kt and the turn rate was about 90 deg in 30 sec, or about Rate 1. This is about 55 HP on the Lycoming O-290 D2.
I then took out the rudder and banked into the turn (about 15 degrees bank was required to maintain Rate 1) to maintain the same rate of turn as the skidding turn but in a coordinated bank. To keep from climbing I had to reduce power. I ended at at about 16" MP @ 1700 RPM, in a banked level Rate 1 turn at 70 kt, which is somewhere around 45 HP.
So it took somewhere around 20% less power to maintain a banked turn at Rate 1 than a skidding turn at Rate 1 when flying near max L/D. The extra drag of the skid was also quite obvious in the deceleration when applying the skid with rudder vs rolling into the coordinated turn.
With airliners, turning by skidding is out of the question because with swept wings you get a powerful roll rate as soon as you yaw a couple degrees and you will quickly run out of opposite aileron trying to keep the wings level.