Neurologists learn to do very detailed exams, studying reflexes, strength, coordination, and many other systems. These involve great skill in noticing important, sometimes subtle, details.
To give you an example, today I was reviewing some notes from a neurologist across town. Here is a detailed complete neuro exam from her note:
Obviously, it takes years of training, and a great deal of time, to notice things like this.
To give you a more thorough picture, here is another very detailed exam by the same doctor on the same patient, done 1 month after the initial visit. Notice how she carefully documents changes in his condition:
So that's what a neurology exam involves.