A lot of considerations will be specific to the specific protein and cell line you're using, and what question you're trying to address. If you're trying to make a lot of protein for purification, overexpression will definitely help you recover more protein with fewer cells; however if the overexpression is too robust, you may have issues with proper protein folding (aggregation) or the proper incorporation of post-translational modifications. If you're looking for physiological activity of a protein in a signaling pathway, or an enzyme in a metabolic pathway, you may get increased signal/metabolism, but you may also exhaust other players in the pathway (i.e., something else will become the limiting factor). The same can be a factor for sub-cellular localization experiments; increased expression would increase your signal, making your protein of interest easier to find microscopically, but you could express too much and overwhelm the trafficking machinery leading to mislocalization.
One overall advantage would be the consistency from experiment to experiment. As opposed to transient transfection, if you're using an overexpression cell line, then you won't have to deal with the day-to-day variability of transfection efficiency.
I would say one of the biggest advantages is the great flexibility in the protein you can express (e.g. testing the effect of different mutations or isoforms). Also the expression level is a big advantage (e.g. recombinant protein production).
One big disadvantage is definitely that it is a highly artificial model with an unphysiologically high expression level. Another problem is that assays can be affected by different transfection efficiencies and one could use cell lines which are unsuitable for the question.
A lot of considerations will be specific to the specific protein and cell line you're using, and what question you're trying to address. If you're trying to make a lot of protein for purification, overexpression will definitely help you recover more protein with fewer cells; however if the overexpression is too robust, you may have issues with proper protein folding (aggregation) or the proper incorporation of post-translational modifications. If you're looking for physiological activity of a protein in a signaling pathway, or an enzyme in a metabolic pathway, you may get increased signal/metabolism, but you may also exhaust other players in the pathway (i.e., something else will become the limiting factor). The same can be a factor for sub-cellular localization experiments; increased expression would increase your signal, making your protein of interest easier to find microscopically, but you could express too much and overwhelm the trafficking machinery leading to mislocalization.
One overall advantage would be the consistency from experiment to experiment. As opposed to transient transfection, if you're using an overexpression cell line, then you won't have to deal with the day-to-day variability of transfection efficiency.