I think Physiatry and Neurology are similar in that we can both see the same type of patients (Stroke, Parkinson's, Alzheimer's, Cerebral Palsy, Polio, Muscular Dystrophy, etc.). In fact, on the inpatient Stroke service, most of our referrals are from the Neurology service. We see these patients at different stages of their recovery. Neurology tends to deal more with the acute management and diagnosis of CNS disorders. Physiatrists tend to focus on their functional recovery (possibly involving botox/phenol/joint injections, gait analysis, orthotic prescriptions, etc.) as well as the medical management and prevention of complications that often ensue. At Harvard, we spend two months on the Neurology service as a PM&R resident. It's not uncommon to admit a CVA at the end of your Neurology rotation and discharge them after three days and then see them on the Physiatry service for six weeks. Another similarity is that we both get trained to do EDX studies. Both specialties also get a lot of training in ordering appropriate imaging studies and interpreting them. The difference is that Neurologists are better at looking at the brain while Physiatrists are better at looking at the MSK system.
Other differences in our training can also lead to very different types of patients. We get much more exposure to musculoskeletal medicine and interventional procedures. I'm not saying that these things aren't available to Neurology residents, but it certainly isn't a standard part of the Neurology curriculum. One of the attendings that we rotate with as a PGY-2 is a former Neurologist (who also completed a Cerebral Vascular Disease Fellowship after his Neuro residency) at HMS. After practicing several years as a Neurologist, he went back and did a PM&R residency at Harvard. He now does Occupational & Musculoskeletal medicine.
PM&R is very broad field so you can really see the same types of patients if you do inpatient rehabilitation or you can see really different types of patients if you decide to do Spine/MSK/Pain/Sports. I've included a link to a
Physiatrist at UWash to give you an idea of what he does in practice.