I am in charge of my subordinates at work. I am able to give them orders, and make decisions regarding what they do, and how they do it.
I am responsible for my children. I must ensure their safety and care.Their behavior is a reflection of the decisions I made while raising them. I must answer for their health and behavior both in the home and outside the home, good or bad.
The two words can be used interchangeably, but "responsible for" implies a level of personal ownership, and may include legal responsibility. "In charge of" implies a legal responsibility, but not necessarily any personal ownership.