What are advantages of each configuration or they may be considered similar? Difference is in waveguide short wall position; quarterwave for equivalent shunt array and half wavelength for equivalent series resistances array. Overall design procedure and characteristics look pretty similar. Weighting is based either on combining slot conductances or resistances to obtain desired radiation pattern and matching.
Coupling between antenna elements would be different when using shunt or series configuration.
Mutual coupling between antenna elements affect the radiation resistances of the coupled elements, affect the side lobes, and affect the main beamwidth.