Is there a hard and fast rule of which one is stronger than the other, or does it depend on their context in a protein?
The mechanical stability of a protein (I am assuming by strength you mean mechanical stability) depends greatly on the general context and there is no clear connection between alpha helix or beta-sheet content and stability. For example fibroin, one of the main components of spider silk and one of the strongest proteins known, consists mainly of beta sheets while keratin another very stable structural protein (the stuff hair and fingernails are made of) consists of four alpha helices alternating with three beta turns is believed to get most of its strength from hydrophobic interactions within its alpha helices. Also, actin and myosin muscle fibers consist mainly of alpha helices. The list goes on and on with no clear pattern which secondary structure forms more stable proteins.