Umbrella term for "parts of the brain"
When you just want to list the components of the brain, that is to say what the brain consists of from the gross anatomy perspective, you can say that the main parts of the brain are:


cerebrum
cerebellum
brainstem


You can also say that the parts of the brain are some smaller, but anatomically well defined components, such as the pineal gland and pituitary gland.    

When you intend to describe the functions of anatomically well defined brain areas, you can say that the brain structures are cerebral lobes, thalamus, hypothalamus, hypocampus, amygdala, putamen, corpus callosum, cyngulate gyrus, etc.

A part is "a portion or division of a whole that is separate or distinct; piece. A structure is "mode of building, construction, or organization; arrangement of parts, elements, or constituents." So, it's not about what "it is," but about how you look at it. A thalamus is a part of brain and is a structure made of several nuclei...  

You can see these terms used very differently in various texts, but a short overview of brain anatomy by Mayfield clinic use them in the same way as I did. These terms are not a part of any official classification in neurology.