I think that the cardinality of E' should always be greater than the cardinality of $E^*$, so they never will be isomorphic in any sense. Basically, as pointed out here, $E^*$ is the space of all maps from a topological basis of E into a field. E' is analogously the space of all maps from an algebraic basis to a field. So, this boils down to the question:
I think the answer is no. For example, in $l^2$ (which is the smallest infinite-dimensional Banach space), the topological basis is countable. As Ady points out, the algebraic basis should have cardinality $2^{|\mathbb N|}$.