This may not be as exciting a post as the one about the internal differences between these two groups, but it has lots of useful tips for any budding paleontologists who want to know if they've found a brachiopod or a bivalve.
All in all, this comes down to symmetry. If you hold a brachiopod in your hand so that you are only looking at one valve of the shell, and then you flip it over to look at the other valve of the shell, you'll notice they aren't the same. Do the same thing with a bivalve, and you'll notice they are. If you were to look at the two valves of a bivalve shell you would see that they are, in fact, mirror images of each other (with a few exceptions, most notably the oysters). This means that the plane of symmetry in a bivalve runs right along the hinge line.
For brachiopods, it's the opposite. The plane of symmetry in brachiopods runs perpendicular to the hinge line; if you cut a brachiopod in half perpendicular to the hinge line, you would see that both halves are mirror images of each other! Cool, huh?