The two phrasal verbs have more or less the same meaning. When something `cools down’ or `cools off’, it becomes cooler than what it was earlier. In other words, the temperature drops a little bit. During summer, when it cools down, things seem a little more pleasant.
*It was well past midnight before it started to cool off/cool down.
The two expressions can be used with people as well. When you say that someone has cooled down/cooled off, what you mean is that he has become less angry.
*There’s no point talking to you now. Go to your room and cool off.