What is the difference between working stress method and limit state method in the design of beams, slabs, columns, and footing with examples that are easy to understand?

The Working, Allowable or Permissible stress method is an elastic design method. In this design method, members are designed to never go beyond their elastic range. The primary benefit of this benefit is that the same loads used for checking flexural, shear, torsional and axial design can also be used for serviceability checks, ie. the loads are not amplified. This reduces book keeping efforts and removes one potential source of error introduction. In practice, this means it is simpler.

Limit State or Load Resistance Factor design uses the ultimate strength of a member, beyond initial yielding, to determine the allowable strength. The primary benefit of the limit state design method is that it is generally produces more economical designs than Working Stress Methods, and it provides a more consistent safety factor across all elements.  LRFD allows for more control and understanding of the structural behavior by explicitly considering each load type independently. 

In practice, the two design approaches have been calibrated to provide similar results for common loading ranges. Things get a bit out of alignment when the dead load to live/wind/earthquake load ratio is very high or low.

There are many articles on this topic if you google "ASD vs. LRFD". These will not only explain in more technical detail, they will also give you the historical background to understand how these two methods came to be.