One thing they don’t mention is that smooth G2/G3 corners will print horribly if added to vertical corners, there just aren’t enough layers even with a 0.2mm nozzle. 3D-printing a macbook air-like shape with good surface finish is nearly impossible.

I wanted to play with this in OpenSCAD; here is G1 vs G2

  include <BOSL2/nurbs.scad>
  $fn=16;
  back(400) cuboid([200,200,100],rounding=50,edges="Z");
  pts=subdivide_path(square([200,200],center=true),8);
  linear_extrude(100) polygon(nurbs_curve(pts,2,splinesteps=$fn/4,type="closed"));

These corners are so close that they're going to have no practical difference when 3D printing them, the maximum deviation between G1 and G3 is only 0.1mm. You need to exaggerate the effect much more to show the difference.

> G3 continuous corners mean that the print head experiences smooth acceleration while printing such corners.

Axial acceleration is the key here, not just acceleration, that however does not matter if the controller does not output feedrate profiles with smooth acceleration to go along with it.

I thought this was going to talk about all the competing, different corners radiuses on MacOS windows

(is plural of radius radiuses? or radii?)

As someone who modeled surfaces like this for a living:

  G0 Positional Continuity: The surfaces touch without gap, but there may be a sharp corner. Example: the corners of a cube  
  
  G1 Tangential Continuity: G0 but additionally the surfaces have the same slope (are tangential) at the point where they touch. Example: adding a circular fillet to the corners of a cube  
  

If we talk about forces (e.g. imagine a skateboard ramp) you go flat (no centripetal force) to circular (constant centripetal force) without any transition inbetween. In effect this will feel like a bump that can throw inexperienced skateboarders of their feet.

This means tangential transitions often do not cut it.

  G2 Continuity: In addition to being G0 and G1 you additionally ensure the curvature is the same where both surfaces meet. This usually means instead of going from a flat surface into a circle you go into a curve that starta out flat and then bends slowly into a radius.  
  

G1 is if the perpendicular lines at the ends of the two curves align. G2 is if the curvature comb at the end of the two lines additionally has the same height (indicating the same curvature at the transition point).

G3 is basically just ensuring that the two curvature combs are tangential at the point where they meet. G4 is ensuring that the curvature combs are not only tangential, but have the same curvature. G5 is taking the curvature of the curvature...

By this point you may be able to sense a pattern.