A clothing rack is the horizontal pole in your closet that you hang your clothes on (but I think it's more common to call this a "clothing rod"). A clothing rack can also be a pole or a circular stand like the ones you find in department stores.
i.e., "She found a cute dress on the clearance rack at Macy's."
A hanger is the hooked, triangular thing you put inside a shirt (or whatever) before you hang it on a clothing rod.
i.e., "I prefer plastic hangers over wire hangers. Wire hangers get bent too easily."
A coat rack is the tall, vertical pole that has hooks on it for your guests' coats, scarves, etc. They're the things people usually keep near their front doors.
i.e., "There's a coat rack near the door, if you don't want to wear your jacket inside."
They're not interchangeable, but they are similar concepts.