Irregularity is a related term of illness.
In countable|lang=en terms the difference between irregularity and illness
is that
irregularity is (countable) a violation of rules while
illness is (countable) an instance of a disease or poor health.
In uncountable|lang=en terms the difference between irregularity and illness
is that
irregularity is (uncountable) the state or condition of being irregular, or the extent to which something is irregular while
illness is (uncountable) a state of bad health or disease.
As nouns the difference between irregularity and illness
is that
irregularity is (countable) an instance of being irregular while
illness is (countable) an instance of a disease or poor health.