Hi Tom,
Snapping is weird/unintuitive, until the coin drops.
Remember, Snap To Nearest Surface is about Snapping Individual Elements!
In Edit Mode, it behaves as expected:

Snap To Same Target will only Snap the Selected Vertices to the Nearest Surfaces of the left Cube.
In Object mode, you’ll have to hold SHIFT while Snapping, in order to do the ‘same’ thing:

I hope this makes sense (you’ll have to try it yourself, of course).