Hello - often it is a matter of how light or reflections fall on and progress across surfaces. Using the CurvatureGraph on a pair of curves is helpful in trying to visualize this - those are shown on the page you linked but it might be more enlightening to fuss with curves yourself with the graph on. The graph shows curvature - if the graph between two curves is un-broken, then there is curvature continuity (G2) there. There can be hard corners in the graph - that indicates the acceleration (rate of change) of the curve(s) curvature is not continuous - that is not G3. G4 is not, I believe going to show in our graph.
The number of points is related to the continuity - Position needs one point in the right place on a curve to be continuous with another - thus a line can be G0 to a curve at each end. Tangency to another curve requires two control points in the right place - the end point plus the next one in. Curvature/G2, needs three points etc. So to be able to control G2 at both ends of am curve independently, six control points is the minimum. For tangency, four control points, and as mentioned for position, two.
it turns out that the simplest curve that can have two points is degree 1, the simplest that can have four points is degree 3 and the simplest that can have six points is degree 5, and so on - degree plus one is the simplest curve that can be made for that point count. That is why BlendCrv makes the degree curves that it does, depending on the continuity requested.
The example images on that web page are perhaps not super helpful since the ‘outer’ curves are all straight lines, i.e. zero curvature so the graphs are all very much the same.
Here’s another crack at it
In all cases the middle curve has increasing curvature as it gets near the ends but the upper two, being curvature continuous with the inputs on either side, drop in curvature again just before the common end points. The lower one does not because it is not trying make that curvature continuous, tangent direction is all it is shooting for…
Dunno if that helps or hinders…