Soft Storey
The soft storey can be defined as in which the lateral stiffness is less than that in the storey above or below that floor. Those elements resisting the lateral force of earthquake incidents in the horizontal (or all possible) direction have lateral stiffness and are considered as total stiffness of the structure.
Weak Storey
A weak storey is the one in which the storey lateral strength is less than that in the storey above. The lateral strength is the cumulative design shear strength of all structural members except unreinforced masonry (URM) infills. The elements which are resisting the force by sharing the lateral storey shear is called storey lateral strength.
Stiffness and strength are not the same as the stiffness is the force needed for a unit displacement and is defined by the slope of the force-displacement graph. Strength is the maximum force that a system can take.
Soft storey refers to stiffness and weak storey refers to strength. And, a soft storey may also be a weak storey if conditions are satisfied.