We have solved the Schrodinger equation with the modified Kratzer plus screened Coulomb potential using the modified factorization method. We have also employed both the Greene-Aldrich approximation scheme and a suitable transformation scheme to obtain the energy eigenvalues equation and its corresponding energy eigenfunctions for CO, NO, and N-2 diatomic molecules. Numerical results of the energy eigenvalues for the selected diatomic molecules have been computed and discussed. The effects of the combined potential parameters on the energy eigenvalues have been evaluated. We have used the energy eigenvalues of the modified Kratzer plus screened Coulomb potential to obtain the vibrational partition functions and other thermodynamic functions for the selected diatomic molecules. Our results are relevant in other areas of theoretical physics and chemistry, and they agree perfectly with the results obtained in the literature.