What are the problems in EXPSPACE \ EXPTIME?
Your argument proves that $\mathsf{NEXPTIME}\subseteq\mathsf{EXPSPACE}$, since if a TM terminates in (nondeterministic) exponential time it cannot write to more than an exponential number of tape cells. 

On the contrary, if a TM uses exponential space it can still run in doubly-exponential time, e.g. a TM that increments a binary counter of $2^n$ bits until wrapping uses exponential space but runs in $\mathcal{O}(2^{2^n})$ steps. 

So the problems that you’re looking for are those that require at most exponential space but whose running time cannot be bounded by an exponential (even though it can be bounded by a double exponential since $\mathsf{EXPSPACE} \subseteq 2 \mathsf{EXPTIME}$). I don’t have a specific example problem in mind though.