They're very different but there is some overlap. Neuro nurses take care of patients with neurological diseases and psychiatric nurses take care of patients with mental illness. I worked in neurology for years and my unit was neurology, neurosurgery and neuro-oncology. Our primary diagnoses were seizures, brain tumors and hydrocephalus. We also saw some really, really rare congenital malformations and degenerative diseases. An adult neurology unit would probably see a lot of stroke, MS, Parkinson's, Alzheimer's (some cross over with psych there) and epilepsy with some other neurodegenerative diseases like Huntington's and ALS thrown in there.
Psych is very diverse and there are a lot of very specific areas you could work in. An eating disorders unit, for example, is quite different from a substance abuse unit. When I was in nursing school
, our clinicals were on one of 3 units in psych: an acute care unit that was primarily patients who'd attempted suicide and people with bipolar disorder or schizophrenia who stopped taking their meds, a substance abuse unit and a woman's unit where the primary diagnoses were Borderline Personality Disorder and PTSD.
Most psych units operate completely differently than your average med/surg specialty unit. Every psych unit I've ever experienced operates as a milieu with multiple groups/day.