This one's for you, Dan!
What are the differences between brachiopods and bivalves, and how do you tell them apart?
The first thing one might notice if looking at them from a taxonomic viewpoint is that they belong in different phyla. Brachiopods are in the phylum Brachiopoda while bivalves are in the phylum Mollusca. This means that bivalves are related to the other molluscs (snails, cuttlefish, squids, nautiloids, octopus, etc.) while brachiopods are related to... brachiopods.
So taxonomically things are looking pretty distinct. On to the insides!
Both brachiopods and bivalves have two major muscles within the shell to keep it closed. Brachiopods have what are called adductor muscles. These muscles contract to keep the two valves closed. Bivalves also have adductor muscles that contract to keep the valves closed. So why do I bring this up as a difference? Bivalves have a second structure that separates them from the brachiopods. This structure is a ligament that joins the valves dorsally. When the adductor muscles in a bivalve relax, the ligament forces the two valves (shells) open. This is why bivalve shells (of dead bivalves) are often found in a "butterfly pattern" where both valves have opened in a pattern reminiscent of butterfly wings.
The absence of a ligament like this in brachiopods means that when brachiopod shells (of dead brachiopods) are found, both valves are often found closed up as though the animal were still alive.
Feeding systems are also completely different between the two groups. Brachiopods feed by means of a lophophore. This structure has a series of ciliated tentacles that can be extended to create a current which allows the brachiopod to catch food particles from seawater.