The question is hard to answer not because of its colloquial character but because it tries to establish a comparison bewteen two predictions at the TeV scale sustained by two very different theoretical frameworks :
- the prediction of supersymmetric particles is made in the context of quantum renormalizable theory with fields interacting in a Minkowski 4D space-time;
- the existence of a fine structure (two-sheets) of spacetime comes with a spectral action principle on an almost-commutative geometric setting.
Despite this fundamental difference, I think it should be possible and interesting to compare them, debate about their mathematical and observational consistency as two effective theories at the TeV scale. I understand effective theories in the modern viewpoint very pedagogically explained by Matthew Schwartz in this lecture note.
I think it's worth emphasizing that the technical naturalness issue of the Standard Model Higgs boson exists only if one assumes that it is embedded in a larger renormalizable theory that goes along the line of conventional QFT (I would appreciate to be corrected if I am wrong on this statement).
Insofar as the spectral action principle, applied on a crude almost-commutative geometry and with a Planck-scale cut-off, already proves to be able to deliver the Einstein-Hilbert and the Standard Model Yang-Mill-Higgs terms, it is not unreasonable to expect that noncommutative geometry offers another embedding of the Standard Model to some kind of UV completion with degrees of freedom that would be different from customary QFT. This recent article(2013) for instance goes along such lines proposing a different noncommutative structure which mixes spin and gauge degrees of freedom in a very specific way.
Nevertheless if one want to stick on the naturalness problem of the electroweak scale it could be interesting to reassess in the light of recent developments the former tentative to look at noncommutative geometry as another alternative to compactification(1999). Indeed a almost-commutative toy model on a two-sheeted spacetime with a gravitational and U(1) gauge fields was proved to give a Randall-Sundrum potential for the Higgs field with the correct exponential term to reduce the natural scale of the electroweak symmetry breaking from the Planck scale to the TeV scale without fine tuning. I quote :
We notice that the non diagonal elements of the matrix algebra, and of the Dirac operator, have a twofold interpretation. On one side they are the Higgs field in the gauge setting, whose natural scale is the electroweak scale. On the other side they appear as the discrete component of the Levi–Civita connection, with a natural gravitational (Planck) scale. It is this dual role which solves the hierarchy problem in this setting
It would be interesting to investigate if the exponential term appearing in the 1999 article has some connection with the new singlet scalar field proposed by Chamseddines and Connes in 2012 to postdict the correct mass of the physical Higgs boson observed at LHC8 ...(beyond the mere coïncidence of the sigma labelling for the new degree of freedom in both papers, just a convenient notation for a generic singlet scalar field I guess)!
Comment about the bounty: The bounty period beeing over, I thank Matt Reece for proposing an answer but I can't accept it for the very reason given by Peter Shor in his comment. I thank also Mitchell Porter to make constructive remarks. I feel free now to propose my own answer.