Easy, schizphrenia is a severe mental illness. OTOH, people who are pictures of mental health have experiences of God all the time.
More generally, the neurological activity of trance states has been shown to be quite different from hallucination. I suggest you look into neurothology if you're interested.
There is a lecture posted up on YouTube. It is about an hour long. It's by Dr. Andy Thomson, a psychologist. He explains neurologically, culturally, socially why we believe in Gods. Search "Why we believe in Gods" and you should find it.
I'll summarize the main points, but obviously in an hour-long lecture, he goes into far greater detail and can explain his own research far better than I can.
He says that we can have something called "decoupled conversations". In your mind you can picture yourself having a conversation with your mother, even if she's thousands of miles away. That image is very realistic. Your brain can build a "model" of your mother and you can engage in a conversation with that image in your head. To primitive people, they would not be able to distinguish a decoupled conversation from a spirit or some other mythological being.
Now take that one step further. A close friend or relative of yours dies. Even though that person is dead, you can still "talk" to that person in your mind. You see how the spirituality is building now? You feel you can still communicate with him, but in reality he's just rotting into the soil, unable to hear your words.
Take that another step further. When primitive people started asking themselves "Where did we come from?", the only reasonable explanation given their knowledge was that some mythological god created them. These decoupled conversations led to a seemingly real interaction with these made up gods.
They then project images of their gods onto naturalistic things in their lives...like waterfalls or trees. They then say they interacted with god by communing with those waterfalls and trees. The only difference between divine interaction and schizophrenia is that schizophrenia is far less socially accepted.
Edit: My apologies lol. I forgot to answer the second part of that.
The supernatural world is relevant because that's where most consider God to "exist", if I dare use that word. If one doesn't believe in the supernatural but believes in a deity, then that deity must exist in the natural world, no? Therefore gathering evidence for the existence for that deity should be a piece of cake.