I strongly believe they are the same (i.e. commute with Riemann-Hilbert) but have no reference or proof for this fact.
My (unprecise) argument is that both of them involve replacing a sheaf on a vector space V by the sheaf on the dual space V^* given by taking a covector, and doing vanishing cycles at the origin with respect to it.
They way you should think about this for l-adic Fourier transform is that the stalks of the l-adic transform at covectors are the hypercohomology of the original sheaf tensored with the Artin-Schreier sheaf for that covector. Since the Artin Scheier sheaf snuffs out anything which is constant along the different values of the covector, it only picks up bizarre jumps, which by C^*-equivariance can only happen at the origin. What else measures bizare jumps along the values of a covector? Vanishing cycles!