It can be a fine distinction, since their is certainly some overlap between the fields.
The short answer is the primary focus- for PM+R, the primary focus is on function, and for neurology, the primary focus is diagnosis and acute management. There is certainly overlap between the two fields, but the breadth of training in PM+R is on focused on function.
As a specific example, for a patient with a stroke, a neurologist may have as their main focus on delivering a "brain attack" protocol, including whether to administer tPA, evaluation of the initial CT, MRI, and MRA scans, evaluating the blood pressure medications to allow for sufficient blood flow, assessing whether the stroke is in evolution, etc. While physiatrists in theory could do all of these things, for most physiatrists this acute management and diagnosis is not the primary focus of our training.
For the physiatrist, however, our primary question is what are the patients long term functional goals, and how do we help the patient reach those functional goals. Will they walk again, and what do they need to do in order to walk? Will they need orthoses, body assisted gait training, etc? Do they have any speech and swallowing deficits, and if so, what do they need to do to either recover these functions or compensate for their loss of function? Do they have pain related to their stroke, and if so what is the differential of their pain, and what the different treatment options for their pain? What about their bladder function? Are they incontinent, do they have bladder spasticity? How do you assess their bladder, and what are the different treatment options for their long term bladder management? Do they have spasticity, and what options exist for the management of their spasticity?
While a neurologist can in theory address many of these functional questions, it is not the primary scope of their training.
The other big question is "besides neuro-rehab, what else do they do in their training?"
Certainly, neurologists have greater breadth in their training for things like movement disorders (like Parkinson's disease), neuromuscular disease (like ALS, although PM+R certainly focused on the functional aspect of neuromuscular disease), medical neurology, etc.
PM+R has as their core areas the functional aspects of stroke, traumatic brain injury, spinal cord injury, sports medicine, burn, cancer, cardiac and pulmonary rehabilitation, musculoskeletal pain, spine pain, chronic pain, pediatric function (cerebral palsy, spina bifida, etc).
Another big area is electrodiagnostics. Both fields are well represented in EMGs and nerve conduction studies. However, for PM+R it is a core requirement, while it tends to be more of an elective in neurology. Most physiatry residents are capable of performing electrodiagnostic consultations immediately following residency, while my impression is that most neurology residents would have to finish a fellowship to have a comparable skill set.
Another area would be neuroimaging. while some PM+R residents are very good at reading brain MRI's following residency, my impression is that most neurology residents have a greater breadth of training in reading brain imaging, and for the most part are superior to PM+R residents in this regard.
For what it is worth, when I was deciding on residency, neurology was probably my second option, and I have tremendous respect for neurologists. Ultimately, my main interest was improving the function of patients, and I thought the PM+R training program would best allow me to achieve that skill set. I also liked the scope of training in PM+R better, since I was more interested in things like sports medicine and osteoarthritis care than I was in things like movement disorders and neuromuscular disease. All that said, I can see why someone would want to be a neurologist- it's a wonderful and endlessly fascinating field.
Hope that helps.
