Migraines are caused by something happening inside the brain; the exact cause is still being researched. But the fact that it is inside the brain (an organ without any direct pain sensors) means that the pain caused by the migraine is not directly sensed in the location of the cause.

It also means that the side effects of a migraine are much more than just pain. Scintillating scotomas, tingling in limbs not explained by it 'falling asleep', nausea, emotional and mental confusion, sensitivity to sounds or scents or lights, these are all common enough. And for some migraines, the side effects are more hindering than the pain.

If you have a killer headache and no other symptoms, you have a killer headache. There's a chance it's a migraine (the chance is greater because it's on one side of your head edit: the fading of the pain in a few hours to a day are also a migraine symptom), but it can be any number of things; consult a doctor for advice. Now, if you have (or had in the last 12 hours) one or more of the other symptoms (the exact ones are different person to person and migraine to migraine) it's probably a migraine and you can treat it with that assumption (migraine pills are a combination of other OTC pain medications).

TLDR: migraines are a headache plus a bunch of side effects, while a headache is just pain that feels like it is coming from your head.