To me, a bookshelf is a single shelf. A bookcase is a set of shelves - i.e. 2 or more shelves.

Yes, what Gary said. A bookshelf is more often called "a shelf" that can be used for books in addition to other things. A bookcase has multiple shelves within one unit.

Thank you, Su.Ki.!

Just another comment here... A 'bookshelf' is a single shelf fixed to the wall. A 'bookcase' may be an entire free-standing piece of furniture, containing several shelves, that can be moved to another location.