4WD vehicles are often part-time 4WD, meaning they can be switched between 4WD and RWD or FWD.
AWD vehicles are full-time AWD because part-time AWD is unnecessary given that the drivetrain delivers power where it's most needed.
The clearer distinction is that a 4WD has a second gearbox (aka a transfer case) that either manually or automatically engages the front wheels, the Jeep Wrangler /TJ being the architype. That means the extra wheels cannot be triggered seamlessly. A 4FW has distinct driving modes whereas an AWD uses clutches to bounce between modes as needed.
Bluey has a couple lines about this distinction. In Australia, even small children have opinions on 4WD v AWD.
AIUI, you lose energy because the power/torque has to go through an extra differential, transfer case, etc, before reaching the rear wheels.
Going from the engine through the transmission to the front wheels is already a 'fixed cost', but the extra mechanics (and their friction loses) to the rear wheels an 'OpEx' that you don't necessarily have to pay if you don't need to.
True for a FWD v RWD car decisions, but in an AWD are not the rear differential and driveshaft still turning whenever the rear wheels are turning? Even the clutch packs should still be rotating even if disengaged. The fact that there is no power through them doesn't mean they aren't generating losses.
It sounds like the savings are more in construction costs than fuel savings.