in general neurologists are better at diagnosing dementias, especially the early onset forms, and anyone under 40 with cognitive problems should probably be seen by a neurologist. management of dementia (as they are largely untreatable) is best done by psychiatrists who tend to take a more systematic approach and are in general better at having conversations with family, helping patients come to terms with diagnosis, how to explain this to grandchildren, power of attorney, testamentary capacity, capacity to consent to medical treatment, end of life care, functional assessment, and management of behavioral and psychological symptoms of dementia with a heavy emphasis of behavioral and environmental interventions.
Management of dementia often has more to do with taking away medications rather than piling them on (e.g. is that statin or antihypertensive really necessary? what about that benzo, or oxybutinin making the patient more confused). The anticholinesterase inhibitors and memantine are very limited in what they do but sometimes they can be modestly helpful and in my area primary care doctors are not routinely using these, while neurologists are probably too liberal with them. Use of antidepressants, stimulants, anticonvulsants, neuroleptics (even though this is strongly discouraged) and benzos (even though this is often seen as a contraindication) may be used sometimes too.
About half the patients I see have a cognitive disorder including pseudodementia, HIV dementia, Alzheimer's, vascular, mixed, TBI, drug-induced dementia, FTD, and so on but this because this is my specialist interest.
But I very frequently do neurological examinations, order labs including special investigations, MRIs, EEGs, and occasionally PET and DaT scans for patients.
I think what you probably don't get yet is that most of what it takes to help patients and families with dementia has nothing to do with the neuroanatomy of brain rot (though it's cool to think about) and more to do with the psychological, social, spiritual, ethical and medicolegal dimensions of care.