What is the difference, if any, between 开心 and 快乐
开心 can be either joyful, delighted or the mood of happiness.   

快乐 is the mood of happiness.  The dictionary defines it as:


  形容称心如意的心情


So, we say 我今天玩得很开心，not 我今天玩得很快乐.  

But they can be synonyms as in 和他在一起很开心/快乐. In this case, both denote the sense of the mood of happiness.