Here's my compare/contrast of the two.
A tensioner is needed to keep the chain tight (but not guitar string tight), and needs to be adjusted as the chain wears and stretches.
But, a chain stretches unevenly which causes the chain to run loose,tight,loose,tight,etc as it spins around the gears.
A manual tensioner has to be adjusted periodically (and correctly) because of this.
Is the chain cold/hot, new/old, loose/tight, etc are all variables that make this tough to get "just right".
A "auto/spring" tensioner should be able keep the chain tight without manual adjustment, and allow for uneven stretching/wear, and temp variations.
Harley's auto tensioner has a spring behind the ramp, that tightens the chain to the "correct" tension.
The ramp's ratchet prevent the ramp from moving backwards, thus maintaining the "correct" tension.
Another set of springs are between the shoe and the ramp, which is supposed to allow for uneven chain stretch.
As the chain spins around the gears the shoe can "untension" slightly when the chain runs tight(er).
I believe this "untension" is also to allow for expansion when the primary heats up to operating temp.
The "Wedge" has a single spring behind the ramp, which does NOT ratchet.
As the chain spins around the gears the ramp can move back and forth to allow the shoe to "tension/untension".
There is a hydraulic valve within the wedge that is supposed to maintain the "correct" tension,
but allow slight movement to allow for uneven chain stretch and temperature variation.