thermodynamics - Can an exothermic (endothermic) process increase (decrease) internal energy?

We classify exothermic and endothermic processes based on enthalpy change, $\Delta H < 0$ and $\Delta H > 0$ respectively.

I assume this translates to $\Delta U < 0$ and $\Delta U > 0$, respectively, for most processes. (where $\Delta H = \Delta U + \Delta (PV)$, in general).

But not all? In which case we might have processes where $\Delta H < 0$ (exothermic) but $\Delta U > 0$? Could you give an example?

Otherwise exo and endo could just be classified based on $\Delta U$ alone instead of $\Delta U + \Delta (PV)$.

Background: I'm just trying to find a motivation for using the concept of enthalpy. I know about the motivation due to additional $PV$ work needed for a given process (hence $U$ not enough to account for all the energy involved). Just wondering if the exo/endo classification is another motivation for using enthalpy?

Side note: We know something like this for $\Delta H$ v $\Delta G$, for classifying spontaneous v non-spontaneous processes. In that case $\Delta H$ is not enough and we need $\Delta G$. Just wondering if a similar thing happens in case of $\Delta U$ v $\Delta H$ for exo/endothermic classification.