MatchSrf changes something
But Match (for the curve) doesn’t change anything 
Matching a curve is different of matching a surface. When matching a surface you need to be aware of two principal curvatures. This explains the change when both (u and v) are not zero. You will get a deviation at the 3rd control point row. The other issue is related to the fact of having rational geometry, as already pointed out. That’s why we usually prevent rational geometry at all. We split circles into 4 Bezier curves (with 6, more accurate 7 cps each). Same for cylinder, spheres (8) and so on.
What are the curvatures?
Hello - see-
-Pascal
That is, positive and negative? Or u / v directions?
Somehow incomprehensible 
Imagine matching two curves. The one who is matched against has a domain (parameter bound) of 0 to 1. If you want to match the other at the start of your other curve, you will need match the curvature at t = 0. This happened with the third controlpoint (I’m shorten this here).
Now if you have a surface you have a second parameter which could be named as s. (Usually they say u and v and not t and s). Now you cannot simply match the controlpoint rows like they are a curve. You need calculate the curvature of s(or v) in. That’s what Gaussian curvature is about, which is the product of both principal curvatures.
Why is Guassian curvature needed or used? Why not just match curvature normal to the edge? If the edges are coincident the curvature along the edge matches between the two surface. If the curvature in two directions matches the curvature in all directions matches, with the possible exception at singularities.
True its not needed. I was just adding the last sentence to explain Pascals link to the gaußian curvature. Yes you need to measure both principal curvatures and match them. Since its multiplicated the Gaußian curvature can be the result of infinite combinations of k1 and k2. Of course you want to match k1 and k2 exactly. However if you express surface curvature as gaußian curvature you will notice that surface curvature is not equal to curve curvature because its is a product of the two pc. One dimension higher . That was the intention.
Pracmatic seen. If you have k1 (from iso curves, or cp curves) and the gaußian curvature(by rhinocommon), you could get k2 without measuring it. I think Rhinocommon does not directly provide k1 and k2.If you do all the math for yourself, you will get k1 and k2 anyway.
You don’t need to explicitly match principal curvatures. If curvature in two different directions are matched and the surfaces are both smooth then the principal curvatures will match. Edit: This only applies along a smooth curve which is coincident to both surfaces, such as a smooth shared edge. It does not apply at isolated points. It’s best if the directions matched are close to orthogonal but they don’t have to be the directions of the principal curvatures. Added: Since the curvature along the coincident curve/edge will match then only the curvature in one other direction needs to match. The preferred direction will be close normal to the curve/edge.
The interactive Rhino command Curvature returns principal curvature at a point on a surface as well as Gaussian and mean curvature.
ok, wasn‘t aware of that
I just coded a surface match functionality by using isocurves at the greville parameters. I wrote a match curve command which flawlessly matched G2. Now whenever the surface was double curved I couldn‘t achieve a G2 at the surface. I obviously had to modify the 3rd controlpoint row a bit. Excatly the same way as the thread owner noticed above. I haven’t worked on that further, but my assumption was that my principal curvatures did not match. Is this wrong? Did you coded a match surface component and do you know what I did wrong? Maybe I could tweak my match surface component then and use it. This is something which would be really helpful for me.
here is a link to my „newest“ match curve
This only applies along along a coincident, smooth curve, not at isolated points. Since the curvature along the coincident curve/edge will match then only the curvature in one other direction needs to match. The preferred direction will be close normal to the curve/edge. I should have made this clear in the post above. I’ll revise that post. My apologies for any resulting confusion.
For a mathematical discussion see Geometrical Criteria to Guarantee Curvature Continuity of Blend Surfaces, Penga and Wolter, Journal of Mechanical Design, March 1992. From the conclusion of that paper:
The LC theorem, however, shows that if the blend is already first order smooth, then it is sufficient that normal curvatures agree in a direction other than the tangent to the linkage curve.
https://pdfs.semanticscholar.org/1e2c/0f2f1ae3f562587bd86b68421141dbef8c13.pdf
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thanks, this is very informative.