What skill a student exemplifies if he used the word "dad" instead of using "father"?
I've usually heard father used more formally (in essays, between a stranger and a child, etc.) and dad more informally (friends talking about their dads, a child calling for his dad).
Generally, father is used more formally and dad is more familiar. There are some that claim that "father" merely has a biological meaning. A "dad" cares for and supports his children.
I don't think that a student's usage of dad or father shows a particular skill; I believe that the usage has more to do with his/her familiarity with his/her father.