"My compliments" is a phrase we use to show appreciation for something. It's like saying, "Good job." A common use is "My compliments to the chef," which is used when we enjoy a meal. "Congratulations" is used when someone receives something good. Perhaps they won an award for their hard work or they found out that they will have a baby soon. When a person gets good news, we can often say, "Congratulations."

Not really. 'My compliments' are simply good wishes. As an expression it sounds rather stiff and insincere, and doesn't correspond in meaning to 'congratulations'.