"The neurological approach," as it pertains to psychology and behavior, is a basic physiological (based on how the human body works) study of the mind that aims to correlate the work of the chemical and electrical impulses in our brain and nervous system to the ways that humans behave. It is not just a mere study of what we feel and think, it is an actual root-cause analysis of the influence of the nervous system and brain on our behavior.
In The Future of the Mind: The Scientific Quest to Understand, Enhance and Empower the Mind, physicist Michio Kaku says,
There is a struggle, then, between different parts of the brain concerning the future, which may have desirable and undesirable outcomes. Ultimately it is the dorsolateral prefrontal cortex that mediates between these and makes the final decisions. Some neurologists have pointed out that this struggle resembles, in a crude way, the dynamics between Freud's ego, id, and superego.
This quote entails that there are several powers at work when behavior manifests. Lots of sources, connections, impulses, and even predictions are put together in the making of one behavioral event. Moreover, there have already been specific locales (such as the aforementioned dorsolateral prefrontal cortex) that have been identified as active participants in the behavioral processes of individuals. In all, behavior is not a mere product of reaction and instinct. There is much more at stake.
Hence, neurobiology is essential in understanding preexisting conditions, potential diseases, and anything related to the nervous system. It even helps to understand why some people are prone to certain behaviors, compared to others. The neurological approach has also been very influential in the field of education because, thanks to brain studies, we can almost sketch a map of how humans can learn and how they can adopt new behaviors more efficiently.
Remember that what is known about the brain and its power is limited in comparison to all there is yet to be discovered. Neurological approach studies have been the trying to close the gap in terms of understanding how the brain truly functions. Therefore, it is through neurological studies that we can obtain enough data and documentation to amplify what we know about cognitive psychology and other cognitive sciences.
As for the other part of the question, personality and behavior are intrinsic. Your personality is a composite of the different behaviors that surface when you are exposed to a variety of environments. It is a combination of nature and nurture because you are born with some inherited traits, and then you adopt, abandon, enhance or transform behaviors depending on what you need them for. That being said, behavior itself is what helps mold someone's personality. This is why they are so intrinsic to your being. If someone does not react to the environment and act on it, there is little to no chance to build the unique personality traits that will define you as a person.
As such, neurobiology studies help predict, sometimes explain, and sometimes justify some behaviors, personality disorders (such as anti-social behavior) and types of personalities. The study of how the brain works and how we react to things is essential to understand what it is like to be human.
Included is a list of the top neuroscientists today. eNotes also has a great study guide on cognitive neuroscience, which can help you answer any questions on the topic.