I couple of days ago, I asked extensively the same question on Stack-exchange (see https://math.stackexchange.com/questions/3592151/riemann-hilbert-correspondence-versus-simpson-correspondence)and go no answer.

Let us assume that X is a connected, smooth complex algebraic variety. Then the Riemann-Hilbert correspondence tells us that the functor which sends a flat connection with regular singularities on a vector bundle of X to its asocciated monodromy representation is an equivalence of categories.

Furthermore, the Simpson correspondence tells us that there is an equivalence of categories between the category of complex representations of the fundamental group of curves and the category of semi-stable Higgs bundles with trivial Chern class.

Its seems to me that the Simpson correspondence should be a consequence of the Riemann-Hilbert correspondence, especially since the category of Higgs bundles is roughly the tangent category of the category of vector bundles. However, based on the number of papers written on this, and the fact that Simpson wrote a ICM note on this, this is clearly not the case. So I guess I must miss what the additional content of the Simpson correspondence is. Could someone help me?