Confusing about what's a neurological illness and a psychiatric illness and Anton's syndrome

I recently came across a syndrome called Anton's syndrome. People who
become biologically blind won't accept they are blind. It confused me
about what is considered a mental illness. This is not a psychiatric
illness. It's a brain illness though. It's secondary to some sort of
brain injury that affects the areas of the brain that process sight and
the damage is sufficient to make the person blind. The syndrome itself
sounds like a delusional state in a way where the person doesn't realise
their blindness. This seems like it could be considered as a psychiatric
illness though without cure. It's a neurological illness but then why
isn't schizophrenia a neurological illness if it too is the result of
brain injury or brain deficit.

Here's the Wiki page
http://en.wikipedia.org/wiki/Anton–Babinski_syndrome

And here's a case report.
http://jmedicalcasereports.com/content/3/1/9028
Madulla, M. el at
Anton's syndrome due to cerebrovascular disease: a case report
Journal of Medical Case Reports 2009, 3:9028doi:10.4076/1752-1947-3-9028

"
> Anton's syndrome describes the condition in which patients deny their
> blindness despite objective evidence of visual loss, and moreover
> confabulate to support their stance. It is a rare extension of
> cortical blindness in which, in addition to the injury to the
> occipital cortex, other cortical centres are also affected, with
> patients typically behaving as if they were sighted.
"

And a little detail on the case.
"
> A computed tomography scan of the patient's brain (Figure 1)
> demonstrated evidence of acute infarction in the right occipital and
> left occipito-parietal lobes, on a background of generalised
> periventricular ischaemia, consistent with a diagnosis of cortical
> blindness. Her receptive dysphasia resolved early in the course of her
> hospitalisation, but during her rehabilitiation she nonetheless
> continued to deny any loss of vision and showed signs of
> confabulation. When asked to comment on the doctor's tie, she was
> quick with an answer, but one that was incorrect. Interestingly,
> towards the end of her admission, she asked a nurse to "light up some
> candles" because she felt the room was dark, suggesting a degree of
> light perception that was not present on admission. She required
> assistance in mobilising safely in view of her visual impairment, and
> required help for most activities of daily living. Although she was,
> for example, unable to see her meals, she would feel for the utensils
> on the tray when it was placed in front of her, and if left to her own
> devices she would start eating the food, but assistance was required
> to help her finish the meals and avoid spillage. Although she would
> deny visual deficit, she would accept such assistance.
"