What's the difference in meaning? When should I use each one?
Well-being represents how well someone is faring as a person. This could be with regards to someone's health, mental health, or the environment in which a person (usually a child) lives in. It doesn't necessarily imply that such a person would be happy, simply that they live in a state that's "in their best interests."
Happiness on the contrary is that fleeting thing which people always desire but may or may not always be in their best well-being. I hope that answers your question.