Supermarket Pharmaceuticals

Has there been shown to be any substantive difference between the effects of a brand-name over the counter medicine (say, Advil) and its generic store brand counterpart?

Does sharing the same active ingredient really mean that they work equally well?

Depends.

Using cheaper ingredients may alter the bio-availability of the drug. Sometimes, there is no appreciable difference between generic drugs and brand-name, but sometimes generic drug companies use cheaper versions of the ingredients which may affect the efficacy.