Yes yes yes, we are all familiar with the basic physics of work and mass and distance and elevation. The endpoints are the same so the work is the same. But you're well aware that's not the point.

This is the point: with the help of your momentum carrying your leg through parts of the pedal stroke (which may be reliant on easily-fatigued muscle groups), stronger and/or fresher muscles can carry more of the load..

This is especially evident when climbing out of the saddle, because you can focus your efforts on the most powerful parts of the stroke - the push down and the pull up - with relatively little consideration for the problematic bottoms and tops.

Sure this isn't a problem with perfect technique and unlimited stamina, but most people don't have those things. This is especially true when out of the saddle, after long climb, when you're fatigued and trying to power through to the top. Round pedal strokes go out the window. Some muscles fatigue more slowly than others, and the fixed drivetrain allows those muscle groups to carry the load better than non-fixed.

Furthermore, given that you can rely on momentum to carry you through certain parts of the pedal stroke, you can rest entire muscle groups by focusing on the other parts of the stroke. The energy expended is very obviously the same, but you can better decide which muscle groups expend that energy.



Example:
Say you can feel your hip flexors are fatiguing, so you want to take the 9:00 to 1:00 part of the pedal stroke easy for a few revolutions. Your quads/glutes (the big, strong muscles) feel fine.

On a freewheel you have two options:
A) Stop pedaling all together, and coast. Your hip flexors do zero work, but until you turn them back on, none of your other (potentially fine-feeling) muscle groups do either. you slow down considerably.
B) Push lightly through that part of the stroke with your hip extensors (or compensate by pulling harder on the equally fatigable hamstrings), then back on the power with your quads. Your hip flexors use less energy than normal, but don't get to rest entirely.

On a fixed gear you can do option B, or option C) Turn off the hip flexors entirely and "coast" through that part of the circle, and still have the pedals get around to 1 o'clock for the power stroke.

The same is true for any other combination of fatigued or fresh muscle groups. To coast on a freewheel robs you of the opportunity to maintain speed by compensating with other muscle groups. To "coast" on a fixed gear lets you work which groups you want.