Not sure exactly how your prof distinguished them, but maybe the first distinction is talking about that entropy does increase even if thermal energy exchanged is zero.
Consider, the case of Free-Expansion, where in an enclosed chamber an ideal gas of $n$ moles is confined to a smaller volume $V_i$ by a partition.
When the gas is allowed to get dispersed over the whole volume $V_f$ by puncturing the partition, it does no work on the partition; neither does any thermal energy come in from outside nor does any get out to outside.
The internal energy remains the same.
It can be easily found out using the First Law and that change in the internal energy is zero (since the gas performed no work and no thermal energy is exchanged),
$$\Delta S_\textrm{free-expansion}~=~ n\mathcal R\ln\frac{V_f}{V_i}~~\gt 0 ~~~~~~~~[\because ~V_f\gt V_i]\,.$$
So, entropy change is positive even though no thermal energy is exchanged.