We report a molecular dynamics (MD) study of the interfacial behavior of key partners involved in the Cs+ cation extraction by a calix[4]arene-crown-6 host (L), comparing an ionic liquid (IL) to a classical molecular solvent (chloroform) as receiving "oil" phase. The IL is composed of hydrophobic 1-butyl-3-methylimidazolium cations (BMI+) and bis(trifluoromethylsulfonyl) imide anions (Tf2N-) and forms a biphasic system with water. The simulations reveal similarities but also interesting differences between the two types of interfaces. Much longer times are needed to "equilibrate" IL systems, compared to classical liquid mixtures, and there is more intersolvent mixing with the IL than with chloroform, especially concerning the water-in-oil content. There is also some excess of the BMI+ cations over the Tf2N-anions in the aqueous phase. Simulations on the Na+NO3- and Cs+NO3-ions show that they sometimes interact at the interface with the IL ions, forming hydrated intimate ion pairs, whereas they are "repelled" by the classical interface. The LCs+ complex and L ligand also behave differently, depending on the "oil phase". They are better solvated by the IL than by chloroform and thus poorly attracted at the IL interface, whereas they adsorb at the chloroform interface, adopting well-defined amphiphilic orientations. The results are discussed in the context of assisted ion transfer and provide a number of arguments explaining the specificity and efficiency of IL based, compared to classical extraction systems.