The words overlap in their semantic field and can be used interchangeably, but the connotations of the two words vary.
The etymology of illness indicates the malevolent nature of the physical condition:
"disease, sickness," 1680s, from ill + -ness. Earlier it meant "bad
moral quality" (c.1500).
c.1200, "morally evil" (other 13c. senses were "malevolent, hurtful,
from Old Norse illr "ill, bad," of unknown origin.
Not considered to be related to evil.
Main modern sense of "sick, unhealthy, unwell" is first recorded
mid-15c., probably related to Old Norse idiom "it is bad to me." Slang
inverted sense of "very good, cool" is 1980s. As a noun, "something
evil," from mid-13c.
The etymology of disease indicates the discomfort and inconvenience of the physical condition:
early 14c., "discomfort, inconvenience,"
from Old French desaise "lack, want; discomfort, distress; trouble,
misfortune; disease, sickness," from des- "without, away" (see dis-) +
aise "ease" (see ease).
[c.1200, "physical comfort, undisturbed state of the body;
tranquility, peace of mind," from Old French aise "comfort, pleasure,
well-being; opportunity," which is of unknown origin, despite attempts
to link it to various Latin verbs; perhaps Celtic. According to OED,
the earliest senses in French appear to be 1. "elbow-room" (from an
11th century Hebrew-French glossary) and 2. "opportunity." This led
Sophus Bugge to suggest an origin in Vulgar Latin asa, a shortened
form of Latin ansa "handle," which could be used in the figurative
sense of "opportunity, occasion," as well as being a possible synonym
for "elbow," because Latin ansatus "furnished with handles" also was
used to mean "having the arms akimbo." OED editors add, "This is not
very satisfactory, but it does not appear that any equally plausible
alternative has yet been proposed."]
Sense of "sickness, illness" in English first recorded late 14c.; the
word still sometimes was used in its literal sense early 17c.
By constant use the malevolent nature and the discomfort of sickness have commingled in the two words illness and disease.
According to Wikipedia:
In many cases, terms such as disease, disorder, morbidity and illness
are used interchangeably.3 There are situations however when
specific terms are considered preferable.
The term disease broadly refers to any condition that impairs
the normal functioning of the body. For this reason, diseases are
associated with dysfunctioning of the body's normal homeostatic
process.4 Commonly, the term disease is used to refer specifically
to infectious diseases, which are clinically evident diseases that
result from the presence of pathogenic microbial agents, including
viruses, bacteria, fungi, protozoa, multicellular organisms, and
aberrant proteins known as prions. An infection that does not and will
not produce clinically evident impairment of normal functioning, such
as the presence of the normal bacteria and yeasts in the gut, or of a
passenger virus, is not considered a disease. By contrast, an
infection that is asymptomatic during its incubation period, but
expected to produce symptoms later, is usually considered a disease.
Non-infectious diseases are all other diseases, including most forms
of cancer, heart disease, and genetic disease.
sickness are generally used as synonyms for disease.5 However, this
term is occasionally used to refer specifically to the patient's
personal experience of his or her disease. In this model, it is
possible for a person to have a disease without being ill (to have an
objectively definable, but asymptomatic, medical condition), and to be
ill without being diseased (such as when a person perceives a normal
experience as a medical condition, or medicalizes a non-disease
situation in his or her life). Illness is often not due to infection,
but a collection of evolved responses—sickness behavior by the
body—that helps clear infection. Such aspects of illness can include
lethargy, depression, anorexia, sleepiness, hyperalgesia, and
inability to concentrate.
Interestingly, the current use of the words, reverses the etymological implications with disease representing the objective nature, and illness representing the subjective experience of a malady.