Work has been a little busy lately, and so I haven't gotten around to the external morphological differences of brachiopods and bivalves yet. For that, I apologize. In the meantime, here are some interesting tidbits quoted in Gould and Calloway's 1980 paper "Clams and brachiopods - ships that pass in the night" (published in Paleobiology v.6, no.4):
Louis Agassiz writes in 1857, "Every zoologist acknowledges the inferiority of the Bryozoa and the Brachiopods when compared with the Lamellibranchiata..." I disagree with Mr. Agassiz, but at least he quantified his statement. He is speaking of zoologists, people who deal in the present. In the present, bivalves (lamellibranchs) are much more common fixtures of the marine realm than brachiopods.
100 years later, Ernst Mayr writes, "Our knowledge of comparative physiology is still so elementary that we do not know, for instance, whether or not the cellular biochemical pathways of the mollusks give them superiority over the brachiopods, as one might suspect from a study of the geological record of these phyla."
What these two scientists fail to realize, and what Gould and Calloway so eloquently point out, is that the bivalves and brachiopods coexisted for quite some time. A major reason the bivalves seem to dominate the Mesozoic and Cenozoic is that the brachiopods took a major hit during the Permian-Triassic extinction. The reasons for this are unclear, but it was this disappearance of brachiopods and opening of new niches that allowed the bivalves to flourish as they did.