Both PM+R and Neurology deal with many of the same issues- neuromuscular disease. Both take care of stroke patients, brain injury patients, cerebral palsy (although both encourage some pediatric specialization), etc. For those areas where the fields overlap, neurologists tend to focus more on diagnosis and medical management, whereas as phyiatrists tend to focus more on multiple modality management (including medication) and restoration of function. The emphasis on function is the unique aspect of PM+R, and in my biased opinion, it's strength.
Neurology have some areas of special focus, including epilepsy, sleep disorders, EEG, movement disorders, multiple sclerosis. While some physiatrists focus on these areas as well, generally the training in a neurology residency will be more comprehensive
Physiatry residencies tend to have a broader scope, focusing also on musculoskeletal training, sports medicine, spinal cord injury, rheumatological disorders, amputee medicine, etc. While neurologists may get some exposure to these areas, it is not in the depth that physiatrists do. Additionally, electrodiagnosis (EMGs and nerve conduction studies) is a core part of PM+R training, while it is rare for the neurology residency to develop competency in electrodiagnosis without fellowship training.
The other big difference is that PM+R training is much more procedurally oriented. In addition to more residency training in electrodiagnostics, residents usually develop strong competency in joint injections, and depending on the program, some competency in Botox injetions, alcohol/phenol blocks, lumbar interventional spine procedures, etc.
Like many physiatrists, I loved neurology, and still find the diagnostic emphasis interesting and valuable. In practice, however, the emphasis on restoration and maximization of function is what led me to physiatry. I am very happy with that decision, and would definitely make the same decision again.