What are the problems in EXPSPACE \ EXPTIME? Based on the diagram below (from Papadimitriou's book) we can see that PSPACE ⊆ EXPTIME ⊆ EXPSPACE. We know that PSPACE ⊊ EXPSPACE, which means that there are problems that can be solved in polynomial bounded space and problems that are only solvable in exponential bounded space. These latter problems are outside PSPACE, so they must be either inside EXPTIME or outside of it. The ones inside are problems that need exponential time to be solved, but not necessarily exponential space. The ones outside are problems that require exponential space to be solved and, since a machine can only do a constant number of movements per transition, they also require exponential time. So my question is: shouldn't this put them all back inside EXPTIME? It doesn't make sense to me that there are problems inside EXSPACE but outside EXPTIME, i.e. problems that require exponential space without requiring exponential time. What are the problems in EXPSPACE outside 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.