They are basically cybernetically enhanced humans. At birth, a demigod is implanted with a Mantra reactor (that thing Yasha pulls out of his chest for Asura), which is what lets them absorb and use mantra in battles. This has the side effect of turning their bodies to metal and giving them a more machine appearance. The change is apparently irreversible, and tied to the amount of Mantra you use. Most demigods, like Asura's wife and daughter, do not ever process the kind of mantra necessary to gain the halo, so they don't undergo the metallic transformation.
Originally the demigods weren't all that separated from humans. They were just their champions, fighting the Ghoma on our behalf. They protected the humans, who made the matra that powered their abilities. It was a mutually beneficial relationship. It wasn't until Deus' coup that they assumed the title of gods and began ruling over humans forcefully. Our mythology is based on legends and barely understood records kept by humans living at the time. No more than stories, passed down thousands of years. The details have become obscured, and of course to your average human a Demigod would appear as an actual divine figure. Especially if that was what they were calling themselves.