Mostly the same, but they’re used slightly differently. 高兴 generally isn’t used to describe long-term happiness (if I wanted to translate the phrase “a happy childhood”, I would probably say 快乐童年). I would use 高兴,or maybe 开心,though, to describe how I feel after getting a good score on an exam, which is a more fleeting sort of pleasure. 快乐 is often more long-term, although i dunno if it’s exactly /wrong/ to use it to describe shorter periods of time. It’s a pattern, but not necessarily a rule.
The biggest different is that in phrases like “happy new year” or “happy birthday” you have to use 快乐 or sometimes 愉快。 for example, 圣诞节快乐,生日快乐