In asymptotic region (high energy $s\rightarrow \infty$ and small momentum transfer $t$), the eikonal approximation means we drop out any diagram that has connections between internal lines; or it is corresponding to taking infinite ladder and cross-ladder Feynman's diagrams (including tree-level diagrams) in calculation of scattering amplitude and differential cross section. So, we will obtain leading contribution to the scattering process, and, by this reason, this is a good approximation.
Born series is an expansion whose terms corresponding to Feynman's diagrams. The first Born approximation is corresponding to tree-level diagrams, while other Born terms is corresponding to higher-order diagrams. So, theoretically, the eikonal result will be compatible with the Born series's result. However, practically, we usually take just the first and second Born approximations. So, the eikonal approximation seems to be better approximation.