It's not very competitive because there aer not that many people who want to go into Neurology, just a passionate few. It does not afford a cush lifestyle of a gigantic paycheck compared to many more competitive specialties.
As for the stereotype that Neuro is full of uncurable diseases, that topics has been beat to a bloody pulp. The funny thing is that most people think the frustrating parts are, "Here's a pretty picture of your brain tumor that we can't do anything about" or "We know you had a stroke, but you're SOL." Actually, those cases are more like, "Here's a picture of your brain tumor. Form my expertise, I believe it is a such-and-such kind of tumor. We will work inconjunction with the neurosurgeons and radiation oncologists about how to arrange surgical, radiation, and chemotherapeutic treatment, as well as talk about rehab options" and "You had a stroke at such-and-such time. We can give you clot busting medications for that/send you to a neurointerventional suite to get the clot lysed or pulled out/ admit you to do more workup. We will find the etiology of your stroke, set up secondary prevention for future strokes, and set up rehab for you."
The really frustrating thing that people don't mention as much is "You may or may not have epilepsy, but your biggest problem is that you have depression, anxiety, and many cluster B personaility traits. The spells you are having where you bark like a dog/start break dancing on the floor/become unresponsive, close your eyes really tight, and start hyperventilating are NOT related to a Neurologic problem. No, really. No, I'm not telling you you're faking it. Yes, I am saying you'r crazy."