nLab's page on full sub-2-categories says the following:
If $C$ and $D$ are ordinary categories regarded as 2-categories, a full sub 2-category $F : C \hookrightarrow D$ is equivalently a full subcategory of $D$.
I'm confused by the word choice. Can't a 2-category always be regarded as a 1-category by forgetting its 1-transfors (its natural transformations, etc.)? So, shouldn't all full sub-2-categories be full subcategories as well?
With a general 2-category, which may have non-identity 2-cells, we have a different notion of full sub-2-category than of full subcategory. The two notions only coincide if the 2-category happens to have come from "promoting" a 1-category, and thus has no non-identity 2-cells.
So, yes, all full sub-2-categories comprise full subcategories as well (when we forget all the 2-cell structure and re-look at everything as 1-categories). But the reverse implication does not hold, because the notion of a full sub-2-category does indeed involve conditions on the 2-cell structure.