I've long wondered about the difference between these two. A chiste is just a humorous story, you know, the whole "Two guys walked into a bar..." thing.
A broma, on the other hand, is more like a practical joke. Like putting a fart cushion in someones seat. However, all bromas are not create equal. You have a "broma inocente" and a "broma pesada". A "broma inocente" is a light-hearted joke, while a broma pesada" would be something a little more serious. More like a cruel prank, something that went a little to far.
There's an interesting discussion about "bromas pesadas" here:
Lastly, you can play jokes "Jugar bromas" or "Hacer bromas".