Sickness is used most often in contexts concerning work and insurance. It is commonly found with words such as pay , leave , absence and insurance . Illness has a wider range of uses and is found in more general contexts.
( rather formal ) the state of being physically ill or having lots of health problems
She resigned because of ill health.
Ill health often lasts a long period of time.
illness or pain
When trouble is used with this meaning, it is necessary to say which part of the body is affected.