Biology is a diverse field of science that is primarily concerned with living organisms and everything associated with living organisms. Microbiology is a sub-field of biology, and is concerned primarily with the study of microorganisms. Though microbiology is a sub-field, it has many sub-fields, such as water microbiology and food microbiology.
One of the primary differences between biology and microbiology is that the study of biology can often be conducted with nothing more than the naked eye, while microbiologists almost always depend on microscopes for their studies. Biologists often use microscopes, but many do not, and use other tools to conduct their research.
Another difference between biology and microbiology is that microbiology is far more specific than biology. Microbiology is a diverse field, but biology encompasses everything from genetics to biomechanics to paleontology. This wide diversity allows biologists to see a larger picture of the living world, while the specificity of microbiology gives a more focused view on one specific area of that natural world.
Biology, as diverse as it is, is concerned with all the different scales of size that organisms exist on, but many fields of biology are concerned with more complex organisms, such as mammals. Microbiology is concerned specifically with smaller, individual organisms. Microbiologists might be concerned with something as big as a bacteria, and they do study systems, such as the immune system, but generally they focus on smaller individual organisms.
The history of the two fields also have some significant differences. Many advances in biology depended on the development of the microscope, which is the primary tool of microbiologists, but biology was studied far before the field of microbiology was ever invented. Hippocrates and Aristotle, both of ancient Greece, were early biologists who studied fields such as medicine and ecosystems. Neither would have self-identified as biologists because the name came much later, but both studied life and the natural world.