Because the title of the thread is "concern about neurology" I think it's important to make a couple of general statements before answering your question. I'm extremely happy with my choice of neurology, despite my comments/warnings about negative aspects of the specialty. If it all were to be done over again, I would still do neurology and a sleep medicine fellowship.
I'm a private practice sleep specialist (somnologist if you like) who sees exclusively sleep medicine patients. Like TN said, subspecialization is one way of insulating yourself so to speak from areas of general neurology that you don't care for. I love sleep patients. I love the pulmonary aspect of what I do. I love narcolepsy cases, and the exotic neuroanatomy involved in sleep/wake. I love seeing people get better. I love knowing that I have made their cardiovascular and neurovascular health just a bit better (refer to my interest in stroke medicine below). I like having diagnostic testing that I am personally responsible for (ie *not* some radiologist) that pertains to patients that I personally see in clinic. I didn't want to take call, see pain patients, see headache patients, manage pseudoseizures, manage depressing intractable neuromuscular diseases, or manage terrible nervous system cancers.
Do I still have tougher patients? Yes. They're called insomniacs and they usually have alot of psychiatric overlap. You'll find that these psychiatric/pain/difficult/vague patients percolate into every specialty of medicine that exists. You just see them with different labels. Ask your gastroenterology fellows about IBS or GI pain. Depending on selected specialty, these patients may have any of a variety of complaints. Just ask your friendly neighborhood co-residents in other specialties. For the record, I see one insomniac per day at most and one per week at best.
My own personal, original interests in neurology was actually for endovascular surgical neuroradiology, followed by vascular neurology (I thought of exclusively practicing stroke in an academic setting) and neurocritical care. I liked neurosurgery, liked very sick, intubated patients, liked avoiding general neurology outpatient clinics, and liked avoiding non-ICU inpatient general neurology. But we are a heterogeneous species in neurology, and I'm quite sure you'll find a variety of loves and dislikes on these fourms if you are patient enough and lucky enough to hear a variety of responses. I even thought briefly of movement disorders, too.
Don't like outpatient medicine? Try neurocritical care. Try being a neurohospitalist. Hate inpatient medicine? Try sleep or pain or neuromuscular or headache or epilepsy. And just have an exclusive outpatient practice. One of the very greatest strengths of neurology is its amazing versatility and we often don't focus on that enough on this forum.
As far as your original question, I see zero chronic pain patients where I am responsible for taking care of their pain. From the summary I typed above I hope I communicated that this is quite hard for the rest of us to predict for you. You have to decide on sub-specialty, private practice versus academics, how large a town you want to be in, and how large a group you want to belong to. You see how this could affect your answer? Also, one needs to stipulate about headache patients. Are you counting those as pain patients, too?
As a private practice general neurologist without fellowship training (or having done a neurophysiology fellowship) I would say that you can expect to see a variable amount of pain patients where you are expected to take care of their pain depending on where you want to practice. Now, you will almost certainly be expected to see headache patients. I would estimate I saw one headache patient per half day of clinic in residency. Remember, headaches can be common, and PCP's can manage alot of the easy stuff themselves, meaning that as a neurologist you might be self-selecting for more difficult headache patients. Can you minimize seeing headache patients? Yes, but it's tricky. You could join a practice that has a fellowship trained headache specialist, or a large group with someone who manages the headache workload as a hobby. But I suspect that as a general neurologist in private practice you should expect to see headaches. It's just part of general neurology.
Now moving onwards to the non-headache pain patients, I reiterate that it's likely going to be variable (but on the whole likely rare) to see them - especially if you tailor your practice to avoid them. For instance, try joining a large group or a group with a pain specialist (even better) and you can minimize your exposure to these people. Your practice will ultimately be what you make of it, so if you don't want chronic pain patients *ever,* then you need to establish that boundary at the beginning of your work. Tell your PCP colleagues that you don't do pain and they need to refer to a rheumatologist or pain specialist in the area. Remember that this likely won't apply to headache patients.
And there's a pitfall on permanently avoiding chronic pain (non-headache) patients, too...you might lose more than just your pain business if another neurologist is willing to manage these problems. PCP's are all about your availability, timely care, quality care, patient happiness, and your own versatility (ie the efficieny of using you for neurologic-specific work). Obviously, this will once again be variable depending on where you are, and what the PCP and neuro cultures are where you go. But just remember, no one can make you see something you don't feel comfortable seeing.
Now, if you're willing to see pain patients then again the answer is variable (seeing a pattern here?). You'll probably see more folks like this in Miami or Vegas than Des Moines, for instance, just by virtue of the particular population in a given area. Pain medications are a popular commodity in some locales.
What about academics? In academics you might get more tough cases. Certainly as a resident. But there's a silver lining. You can take advantage of more highly developed ancillary services such as neurosurgery, pain, physiatry, psychiatry, etc that some private practice people can only dream of. And please note that once again, this is dependent on the type and size of academic center you go to. The snazzy places have larger faculty populations and greater and more exotic subspecialization. Smaller teaching hospitals might have *you* as a pain doctor whether you want to be or not.
In summary, in private practice general neurology in a suburb of a large metropolitan area, I bet you can expect to see a headache patient at least once every half day, and chronic (non-headache) pain patients much, much less (if at all), and that you could likely completely avoid the non-headache people if you made a concerted effort to set up your practice this way. Perhaps others could chime in with experience?