I believe that the reason for the discrepancy is that the shader has a number of multiply nodes for scaling that ultimately go into the displacement input, but they're scalar multiplications and the displacement input is vector. This means that the multiplications are all biased by the (1,1,1) vector (i.e., shifted by a scalar multiple of that), whereas the displacement modifier applied to a monochrome map is scaling based on a scalar multiple of (0,0,1), i.e., the z-axis only.
If the input to the displacement shader output is corrected to act on only the z-axis, these views will probably be identical.
That is correct.
In earlier versions of Blender, the Displacement socket in the Material Output Node was grey, taking a single Value (scalar).
Nowadays (since 2.80) it expects a Vector and for that we have a Vector > Displacement Node (not to be confused with the Vector > Vector Displacement Node). This will make the Displacement behave like it used to, only in the Normal direction.
I am not sure if that is what you mean by the discrepancy, as you didn't add a timestamp, but I think it might.
Ah, I see.
Well that difference is in my opinion too small to be caused by the former explanation (but I'd have to watch the whole video and follow along to be sure).
I'd expect that to be caused by the Strength of the Modifier as Kent suspected and maybe the Midlevel or both....