The reactions during the discharge of a dry cell are:
At anode: Zn → Zn2+ + 2e-
At cathode: MnO2 + NH4+ + e- → MnO(OH) + NH3
The difference between a Dry cell and a Mercury cell are as follows:
A dry cell has a very short life span due to the conversion of zinc to zinc chloride that makes the zinc casing porous, while in the case of a mercury cell, the overall reaction does not involve the formation of an ion in the solution whose concentration can change during its lifetime.
In a Dry cell, we use Zn as an anode while in a Mercury cell we use Zn-amalgam as an anode.