Natural candidates for NP-E and E-NP
TQBF (True Quantified Boolean Formulas) is in E and won't be in NP unless NP = PSPACE.

A language in NP-E is trickier. Such a language would also be in NP-NTIME(n) and we don't have great examples of those. You could use a succinct representation like 

$ L = \{ C \ |\ $Circuit $C$ describes a graph $G$ on $|C|^5$ nodes where $G$ is 3-colorable$\}$.

$L$ is in NP, unlikely to be in $E$ but not that natural.