When you think about it, 90% of your body is really just a life support and locomotion system for your brain and the sense organs connected to it. Neurobiology and neurosciences are therefore the branches of medicine that look at the really important stuff.
Between neurobiology and neurosciences, the latter is the wider field and involves computer science, linguistics, psychology, and other disciplines in addition to the purely biological aspects of the brain. Because of this, neurosciences have several subdivisions. Some neuroscientists study how the brain (and disorders thereof) affect human behavior, how this mysterious organ works at a structural, electrical, or molecular level, and even how our brain function affects society as a whole.
Neurobiology has a narrower, medical focus but also overlaps with statistics, social sciences, and other fields of study. Naturally, it makes an excellent preparation for medical school, though a significant proportion of students end up working in other scientific fields.
Neither neurobiology nor neurosciences are degrees you can expect to coast through. In addition to related subjects, theories and concepts related to the brain and its workings are treated in detail. There is also a substantial practical component, possibly including experiments on animals.
Clearly, neurobiology and neurosciences students are attracted to science: curious and eager to understand complex issues at a fundamental level. The ability to apply the scientific method and critical thinking are also very valuable.
Especially if planning postgraduate work after your bachelor's, it's a good idea to select a university that has a significant number of faculty engaged in relevant research. In Europe, this generally means looking at the United Kingdom and Germany, though several other countries have excellent courses taught in English.