Responsibility can mean so many different things. It can mean "You get the blame if this goes south", or "You're going to have to carry this all by yourself.". 

The way I see it, responsibility is a way to get impact. If you have responsibility, you're able to impact. You have the opportunity to make your mark and to do what needs to be done. It also means that you can ask your personal advisory board for help, and grow from their experience. 

To have responsibility is to take initiative, and to lead.