"Being responsible for" something means that you will be rewarded if it's successful, and blamed if it fails.

If something is your responsibility, it means that it's your job to make sure that it's OK. It's your fault if something goes wrong.

We talk about responsibility:

Raising a child is the parents' responsibility, not the school's.

When I was growing up on the farm, it was my responsiblity to feed the chickens.