For example: what makes a neuron in the hippocampus of the brain different from a neuron in, say, the amygdala, or the frontal lobe, or anywhere else in the brain? Do neurons in different parts of the brain have different structures, or are they differentiated solely by location? By "different structure", I mean things like receptors that other neurons don't have and things like that.
$\begingroup$ There are differences in receptor expression and morphology, but it would be beyond the scope of an answer in this site's format to enumerate. The major differentiating factor is most likely the source of their inputs. $\endgroup$– vkehayasJun 30, 2019 at 20:07
$\begingroup$ They express different genes as pointed out by @vkehayas. Is your question about why there is a difference or how the difference came to be? $\endgroup$– WYSIWYGJul 1, 2019 at 11:41
During development of the brain neurons undergo several steps of differentiation. They transition from neuronal progenitors to mature neurons, resulting in different types of neurons, characterized most prominently by their primary neurotransmitter. During the differentiation, different genes are expressed, which in research is used to label specific cell types (to e.g. observe their distribution in the brain). Since timing is important in development and neurons migrate to their side of action, later born neurons often display different features than earlier ones. The final differentiation is also dependent on the environment, meaning surrounding cells can influence the outcome.
Not just neurotransmitters, also other surface molecules, types of synapses formed, morphology and many other genetic differences influence neuron function and their diversity. Each brain area has many different neuronal cell types, some are similar between brain areas, while others might be very specific to one area. As an example, there is a microRNA, expressed in only 6 neurons of the nematode worm C. elegans, that is essential for their specific function to sense CO2 by repressing other genes (publication).
Taking the example of your question, here is a paper on the diversity of GABAergic neurons in the amygdala (there are several different types of neurons having the same neurotransmitter but expressing otherwise specific markers, which results in different specific functions in the circuits). The author compares them to GABAergic neurons in the cortex (including hippocampus), where a similar function might be taken over by a different cell type (expression profiles are distinct due to different lineages and environments and can be labeled by different markers).