We compare the stability of various structures of high coverage self-assembled monolayers (SAMs) of short alkylthiolates, S(CH(2))(n-1)CH(3) (= C(n)), on Ag(111) and Au(111). We employ (i) the ab initio thermodynamics approach based on density functional theory (DFT) calculations, to compare the stability of SAMs of C(1) (with coverages Θ = 3/7 and 1/3) on both substrates, and (ii) a set of pairwise interatomic potentials derived from second-order Møller-Plesset (MP2) perturbation theory calculations, to estimate the role of chain-chain (Ch-Ch) interactions in the structure and stability of SAMs of longer chain alkylthiolates. For C(1)/Ag(111) (C(1)/Au(111)) the SAM with Θ = 3/7 is more (less) stable than for Θ = 1/3 in a wide range of temperatures and pressures in line with experiments. In addition, for the molecular densities of SAMs corresponding to Θ = 3/7 and 1/3, the MP2-based Ch-Ch interaction potential also predicts the different chain orientations observed experimentally in SAMs of alkylthiolates on Ag(111) and Au(111). Thus, for short length alkylthiolates, a simple model based on first principles calculations that separately accounts for molecule-surface (M-S) and Ch-Ch interactions succeeds in predicting the main structural differences between the full coverage SAMs usually observed experimentally on Ag(111) and Au(111).