How would someone use a Psych/Neurology dual-residency career wise?

With the caveat I am a fourth year student who hasn't done the combo residency, here is my understanding (someone else please correct me if I'm wrong):
There are many illnesses that straddle the border of neurology and psychiatry (like Alzheimer's disease, Huntington's disease, Tourette's syndrome, as well as things like pseudoseizures), so if you had training in both specialties you could try to tailor your practice towards seeing more of those kinds of patients. In my rotations, I was actually pretty surprised at how often patients that neurology was consulted on turned out to be having psychiatric issues with no apparent neurologic lesion, so a neurologist with psych training might have an advantage just in dealing with everyday neuro patients for that reason.
However, the impression I get is that a lot of the combined residency programs are being phased out because of the recent creation of the fellowship of Behavioral Neurology/Neuropsychiatry that is open to people who have completed either a neurology or psychiatry residency. If you are interested in a combination of the fields, I think doing a psychiatry residency and going for the neuropsychiatry fellowship would probably get you where you want to be faster. However, I admit that I don't have experience with a combined residency myself, so maybe someone else can provide more info about why you might want to do it.
Good luck with your decisions about school!

I know several med/psych as well as neuro/psych docs. The neurology/psychiatry combo is helpful when dealing with things like dementia, seizures, head injury, sleep disorders. It's much easier to have a combo neurology/psychiatry practice than a med/psych practice.