Abilities are the activities, talents, gifts, tangible or intangible skills you are really good at, such as effective communication skills, verbal and/or written, critical thinking skills, great organizational skills, multitasking ability, etc. Examples of talents would be the ability to sing well, empower others to succeed, play an instrument, bake well, play sports, etc.
Responsibilities are the tasks on your job description (not that you will do them, they are often things we are obligated to do) or your career accomplishments such as you may be responsible for administering an annual departmental budget, or meeting production deadlines, or supervising volunteers on a project, or protecting others from harm.