区别就是两者都是动词+副词短语，且都可表示冷静的意思，具体的不同如下

cool down中文意思是冷却；平静下来，降温

calm down中文意思是平静下来；镇定下来，冷静下来

The passengers often say that seeing animals makes them feel much better and helps them to calm down before a flight.乘客们经常说，看到动物让他们感觉好多了，这可以帮助他们在飞行前冷静下来。