Yes and no. The meanings are the same, but "chill out" is informal and probably falls into slang vocabulary. "Clam down" can be used at any level of formality.

Hi, AlvinC! Although I agree with Kegan that “chill out” is more informal than “calm down” and sometimes they are interchangeable, I think there are used in slightly different situations. First, let me give you a few examples: 1)Calm down, there is nothing to worry about. 2)The wind is calming down. 1) Before we start to discuss it, we all have to chill out. 2) You need to chill out and let go sometimes. or I’m just going to go home and chill out for a bit. In the examples 1) they are interchangeable. In the examples 2) they are not. “calm down” literally means “become quiet and calm, especially after a state of excitement or disturbance” “chill out” = 1)“cool down” 2) relax (chill out area, chill out music)