We consider a conformable fractional Sturm-Liouville problem with the discontinuous(or jump) condition inside the interval. The asymptotic formulas for the eigenvalues, nodal parameters (nodal points and nodal lengths) of this problem are calculated by the modified Prüfer substitutions. Also, using these asymptotic formulas, an explicit formula for the potential functions is given. After all, we discuss Lipschitz stability for the considered problem.