csaps vs. tpaps surface smoothing

Hello, I want to find a smooth function f: RxR ->R describing my data { (x_i,y_i), z_i }_i. For this purpose I am using the functions tpaps and csaps. I'm wondering what is the difference between these methods in the surface smoothing case (by which I mean function input is 2D, output 1D) ? It seems to me that they are minimizing the same energy functional, so the two methods should yield the same results (which they do not). My conjecture is they are minimizing the functional in different function spaces ( some Sobolev space for tpaps and something like C^2 (twice continuously differentiable functions) for csaps.

The function space is unfortunately not mentioned in the documentation and there are no references.

Best regards!