The main reason is that different order curves behave differently when control points are moved. Moving a control point on a higher order curve affects more of the curve than moving a point on a lower order curve. Thus using lower order curves itās easier to create more ālocalā curvature variances, while higher order curves have a tendency to stay āsmootherā. (thatās really a nutshell explanation)

āAppropriateā curve degree will depend on the user and the form to be generatedā¦ Thereās no general rule.

Well, they most likely meant degree 5 and not 5th order, is my guess. Just to tag on to Mitchās comments, degree 5 curves are nice and smooth internally, and still tractable when point editing- but, one additional reason for degree 5 is that the minimum number of points in a degree 5 curve or surface is 6 - which means that on one of these single span curves, you can set and maintain curvature at each end of the curve (uses 3 points) without affecting the curvature at the other end.
Degree 3, also common, are similarly useful with a four-point minimum, for establishing tangency at each end independently.

Iām told by the bigger brains that odd-degree curves are mathematically more well behaved than even-degree ones, but in practice, I donāt know that it makes a lot of difference.

I was working on a set of surfaces that was causing problems and posted in another topic.

One of the problems I have is getting the surfaces to have a smooth transition.

Here I have a set of surfaces. The gray surfaces are fixed and I am trying to fill in the oddly shaped red area. I need to be tangent to the gray at the right but just coincident to the one at the top.

The red surfaces are created with Networksrf with the edges tangent.

Here I have done fitsrf with degree 3. You can see a noticeable crease at where the top and right red surfaces meet.

Here, I have done fitsrf but with degree 7. Notice that the crease has gone away.

As you one would expect, it takes a lot longer to do fitsrf on a 7, 9, 11 degree surface than a 3rd degree surfaceāmuch larger matrices that have to be inverted.