I think you become an adult when you acknowledge the responsibilities of your actions and your everyday decisions.
I make this statement because I've been asking this question to myself many times. It is really difficult for me to try to draw the line in this matter; you could say that adulthood has to do with acting good or bad, but, if so, we would be shifting from adults to childs from one minute to the other because we always behave in a good or a bad way depending on the circumstances and the times we are going through.
After a lot of thinking, I realized that the only difference between myself now and myself when I was a child is that now I am aware of what my actions cause.
When I was a kid, I used to do what I felt like and didn't realize much about consequences, or if someone got hurt by my actions. Today I try to follow my heart in doing what I like to do, but I always consider if I am hurting or upsetting someone.
I think this is the key for adulthood, making choices in total acknowledge of the consequences, which means being responsible for them.