I found a great explanation of the distinction, taken from Kathy Lynn’s book Who’s In Charge Anyway?:

Discipline is not about pain or punishment, nor about revenge or retribution.  Discipline is about teaching, guiding and training.  When we discipline children, we are teaching them the difference between right and wrong.  We’re helping them to learn the consequences of their actions.  They learn why rules exist and how breaking the rules impacts others as well as themselves.  They slowly internalize the information so they can behave appropriately in the future.

Punishment is about causing pain or discomfort in an effort to change behaviour.  We typically think of punishment as being physical — smacking or spanking.  However, threatening, insulting, frightening or humiliating children is also punishment.  And it works only as long as a child is afraid of his parents.  All the motivation is external.  Children learn that parents will hurt them if they don’t follow the rules, but they don’t learn why those rules exist.  And the do not learn how to take responsibility for the real consequences of the misbehaviour.  Instead, they learn to be sneaky, so they won’t get caught, and that they can misbehave when nobody’s watching.  They learn to avoid or disregard pain.  And they learn that hurting someone smaller is an appropriate response to anger or frustration.

Teaching, guiding and training…these are the ways that we learn best.  Punishment might seem to be effective in the short term, but it damages everything from relationships to self-esteem to personal responsibility in the long run.  Discipline might not always be comfortable – this isn’t at all about letting kids get away with everything and suffering no consequences – but it should always be focused on teaching, not pain.