  Variational and perturbative relativistic energies are computed and compared
for two-electron atoms and molecules with low nuclear charge numbers. In
general, good agreement of the two approaches is observed. Remaining deviations
can be attributed to higher-order relativistic, also called non-radiative
quantum electrodynamics (QED), corrections of the perturbative approach that
are automatically included in the variational solution of the no-pair
Dirac$-$Coulomb$-$Breit (DCB) equation to all orders of the $\alpha$
fine-structure constant. The analysis of the polynomial $\alpha$ dependence of
the DCB energy makes it possible to determine the leading-order relativistic
correction to the non-relativistic energy to high precision without
regularization. Contributions from the Breit$-$Pauli Hamiltonian, for which
expectation values converge slowly due the singular terms, are implicitly
included in the variational procedure. The $\alpha$ dependence of the no-pair
DCB energy shows that the higher-order ($\alpha^4 E_\mathrm{h}$) non-radiative
QED correction is 5 % of the leading-order ($\alpha^3 E_\mathrm{h}$)
non-radiative QED correction for $Z=2$ (He), but it is 40 % already for $Z=4$
(Be$^{2+}$), which indicates that resummation provided by the variational
procedure is important already for intermediate nuclear charge numbers.
