What is the difference between a bookshelf and a bookcase?

A book case has a back and a bookshelf does not .

@YARNLADY So I want two book cases. Thanks. I might look on WayFair later for book cases on sale.

A book shelf is something you might hang on a wall and has one shelf whereas a book case is a self contained, free standing unit that may have two, three, four, or more shelves, and even a top area where you could put other stuff.

The clue is in the word shelf, which is singular. A single plank will suffice.

Just a guess:
Bookshelf: the wall is serving as the back.
Bookcase: whether it has a back or not it’s portable. But usage-wise, no difference.