"Prosperous career and happy and healthy family" means that a person or a family has achieved good achievements and balance in both career and family, and it is a beautiful vision and yearning for life.
"Prosperous career, happy and healthy family" means that a person takes care of both career and family, not only succeeds in his work, but also lives happily in his family. This state of life gives people satisfaction and pride, and at the same time encourages people to constantly strive for better achievements and quality of life in their work and life.