The word disease is more specific - it is the medical term for when the human body is not functioning correctly due to infection, genetic defects, or other problems. Cancer, AIDS, and tuberculosis are all examples of diseases. The signs of a disease are called symptoms.
Some common collocations with disease include:
a curable disease = can be cured
a preventable disease = can be prevented
a rare disease = not a common disease
a degenerative disease
a communicable/contagious disease = can pass from person to person
heart/liver disease = a disease affecting the heart or liver
an outbreak of disease
the disease spreading = when more and more people get the disease, in an increasingly large area
The word illness is more general - it describes the condition of poor health, but it is not specific about what is causing the health problems.
Some common collocations with illness:
mental illness = general term when someone has a mental problem. Within the general category of "mental illness," we have specific diseases, like schizophrenia
serious/severe illness = an illness that is very bad
minor illness = an illness that is not very bad
chronic illness = an illness that does not go away
fatal/terminal illness = an illness that will kill the person
recover from an illness = get better after being sick
We also have a few collocations that are used with both:
A diet high in dietary fiber is beneficial to the health of your heart and gastrointestinal tract. Eight raw kumquats provide 9.9 g of dietary fiber. The amount of fiber recommended daily by the Institute of Medicine is 25 g for women and 38 g for men. Dietary fiber also controls blood glucose and insulin concentrations and reduces the risk of developing Type 2 diabetes.