If the intention is about importance, or urgency, or a demand, then you use the subjunctive. Look at it as a kind of imperative, but with a subject. :)
When you use indicative ("...that he works hard"), then you are talking about an action already in reality. Subjunctive actions don't exist in reality, but they are strongly wished-for.
That said, it sometimes sounds formal and even archaic. Still, it's important to know how it's used because we do notice if it's wrongly used or missing! :)