Hierarchy theorems are fundamental tools. A good number of them was collected in an earlier question (see What hierarchies and/or hierarchy theorems do you know?). Some complexity class separations directly follow from hierarchy theorems. Examples of such well known separations: $L\neq PSPACE$, $P\neq EXP$, $NP\neq NEXP$, $PSPACE\neq EXPSPACE$.
However, not every separation follows from a hierarchy theorem. A very simple example is $NP\neq E$. Even though we do not know if any of them contains the other, they are still different, because $NP$ is closed with respect to polynomial transformations, while $E$ is not.
Which are some deeper, unconditional, non-relativized complexity class separations for uniform classes that do not directly follow from some hierarchy theorem?