First, star anise and aniseed (or anise) are not related botanically. Well, they are, but only very far back in the gene pool; both are in the magnolia class. Star anise (which is, confusingly, sometimes called star aniseed) is the fruit of Illicium verum, native to China. Aniseed (or anise) is the fruit of Pimpinella anisum, native to the eastern Mediterranean.
Second, star anise and aniseed both contain anethole, a chemical compound that accounts for their licorice flavor, but star anise has a stronger and more pungent taste.