My Jan saw both, in the time when they were attempting a diagnosis.
The neurologist reckoned she was depressed. The psychiatrist was so useless that he had problems even getting the right notes for the case in front of him.
We also had a psychologist who gave Jan some airy-fairy music to help relax her.
They were all part of the private medicine gravy chain, and each extracted their bit of money before passing us on to their mates.
One might almost wonder whether they know it will be dementia all the time, but have an agreement that, since nothing can be done, and since private insurance won't pay out after diagnosis of dementia, they will simply make hay while the sun shines - for them - so to speak.
The private insurance came with my job, so we used it in desperation as much as possible to get past NHS delays in getting appointments and referrals.
To get back to your question - possibly psychiatrist because, in the absence of any solution organically, some of the experience of handling behavioural problems is all they can think of?
It ia [clearly] all a mystery to me...