Because your mother is a close relative, her DNA differs from yours in chunks. The differences between your DNA and that of someone who is not related to you is more evenly distributed. Working with a few thousand chunks of DNA is much simpler and faster than working with billions of individual DNA elements. A researcher needs only to identify the chunks, and determine which are present in all the relatives with the disease. The gene for the disease should be found in one of these chunks.
Researchers can then use various other techniques to locate the gene more precisely within a chunk.
Each of these groups has advantages and disadvantages. Some of the advantages and disadvantages are practical. For example, when a disease strikes late in life, as does Alzheimer's disease, it will be very difficult to find living parents of cases. When potential control subjects are plentiful, mathematical calculations can help predict what the best group of controls will be for a particular disease. Comparing the DNA of cases and controls is a mathematically and computationally demanding task, which would be nearly impossible without the use of computers.