Confession: For a very (very) long time, we believed that, like soft shell crabs, soft shell clams really did have soft edible shells. Turns out...not quite! Although their shells are comparatively thin and brittle, there's a bigger difference between soft shell and hard shell clams...
All clams feed by filtering sea water through a siphon. In hard shell clams, this siphon is relatively small and short, allowing the clam to close its shell. Hard shells are the most common type of clam sold commercially. Look for little neck, cherry stone, top neck, and quahog clams.
Soft shell clams tend to bury themselves deeper in the sand, necessitating a longer siphon. The longer siphon means that it cannot be fully retracted into the shell and so the shell can never close completely. Whether they're called "soft" shell clams because of their fragile shells or because of this inability to close is anybody's guess! Some more common types of soft shell clams are steamers (popular for New England clam bakes), long necks, and the infamous geoduck.
As far as we understand, there is no significant difference between these two varieties of clam in terms of base flavor. Both clams can also be used for similar preparations, but the siphon on soft shell clams can become quite tough. You can either remove the siphon completely or you can gently slit the surface of the siphon and pull off the tough outer skin.
Related: How to Clean the Sand Out of Clams