Sump:
An open (to atmosphere) external container that contains filter media and a return pump...
Wet/Dry:
A theory of filtration which allows water to trickle over certain types of 'bio media'. This allows the bacteria to at times be in contact with air (higher oxygen content) and at other times be in contact with water (higher nutrient content).
This, at least theoretically, gives the bacteria the "best of both worlds"...
In summary... the sump is the container... the Wet/Dry is the way it is set up...