One pair of curves I had in mind and was referring to is yours, from 5 years ago, in http://www.hammockforums.net/forum/showpost.php?p=22382 .
If I am correct in reading the distances after scaling the y-scale, some of the differences approach a half inch if the depth of the curve is 6 inches.
BTW, the location in the Cloud of this alternative calculator will be of interest to some here,
The proof given in the Shailish Shirali article you cited is that under the realized condition that the suspended mass of the uniform road bed (or body) vastly swamps the negligible mass of the cable (or fabric), the curve is a parabola not a catenary curve. So, the matter is not the difference in curves, but that in practice the curve is never catenary and just should be called what it is, a parabola.
On method, wrote Tee Dee in that same thread:
They are parabolic - think suspended-deck suspension bridge with the hammock occupant the road bed. The threads of the fabric act as the vertical cables on a suspended-deck suspension bridge. You can go ahead and use the cat cuts, but I don't think it will support you as well as the parabolic arcs.
The parabolic arcs are lots easier to make than the catenary arcs. You don't need to make all those measurements and connect the dots.
Just get a good flexible rod, fiberglass or carbon fiber or AL tent poles work very well. I used fiberglass chimney brush handles, 2 4' sections coupled together to make an 8' rod. You can get them at Lowes for about $4 per section.
Mark out the end points, fix the rod at the end points and then flex the rod to the desired shape. Use a marker of some sort, a Sharpie works good, to trace the rod on the material and cut and you are done.
A great parabolic arc.
Note that the head end of the arc is steeper than the foot end to accommodate the extra weight there. You have to adjust the center point where you pull the rod to get that.