"Urgent" is used to describe something that really needs to be done immediately. "Prompt" describes someone who is on time or something that was done in a timely fashion. The potentially confusing overlap between the two comes with their adverbial forms. If something needs to be done "urgently" or "promptly", the difference is very subtle. "Urgently" implies that the action it defines is extremely important and that there could be serious negative consequences if the action doesn't happen.