The difference between them:
Dystopian books are concerned with societies that have gone wrong, or which were bad to begin with. 1984 is a dystopian novel, because it’s about a version of our world where a politically corrupt and all-powerful Party has taken over the country. The Handmaid’s Tale is a dystopian novel, because it is about an oppressive religious regime taking power in America.
Post-apocalyptic books are concerned with the consequences of the end of the world as we know it. The apocalypse could be world-ending, as in a nuclear war (The Road), a zombie apocalypse (Word War Z), a viral outbreak (Station Eleven), or an economic collapse (Soft Apocalypse). In a post-apocalyptic book, human civilization may be ended (On the Beach), or it may go on in a different form (Earth Abides).
To complicate the issue, some dystopian novels are also post-apocalyptic: the apocalyptic event is often an inciting event for the dystopian turn in society (The Hunger Games).