Well, what you've been told is a little off the mark. A headache is a headache--it is the prescence of headpain. A migraine is a little trickier but not necessarily always worse. Migraines can be difficult to diagnose, but they are generally characterized by "migraine-like symptoms." These include aura (visual or otherwise), lethargy, lightheadedness, nausea or vomiting, muscular pain, parakinesia, anxiety, etc......the main idea is that it's usually a headache accompanied by some of these other symptoms. Though many of us have experienced headache-free migraines: sometimes experiencing aura or confusion with other symptoms without a spike of (or the prescence of) headpain.
It's a kind of fuzzy definition because of how little we know about how migraines originate. So, instead, we try to judge as best as we can with (often subjective) symptoms.