While those might mean the same for the laymen, from a medical point of view, there is a difference between illness and sickness.
Medical sociology has long made the distinction between illness and sickness.
Illness is the objective diagnosis that an external impartial observer is able to make based on the constellation of symptoms which the patient presents.
Sickness is the social role that the patient adopts as the patient and other concerned stakeholders, in relationship with the patient, interpret the meaning of the illness.
From what I get of it, someone might see themselves as sick (with the social/role aspect of it) but not actually be ill (in a medical sense).
Also, this paper might provide some useful reading.
ell.stackexchange.com/users/24/Renan
From a British perspective, I'm ill is more common and general term for when you're unwell.
Being sick can refer to actually throwing up or vomiting, but it can also be used for being generally unwell.
ell.stackexchange.com/users/35/Hugo
The formal range of meanings for each word is more or less the same, but they carry different connotations and usage.
It may vary from region to region, but in the USA, it is fairly common to use ill for longer or more serious issues, like cancer, and sick for more immediate things, like the nausea involved in cancer treatment.
Additionally, sick is used in some idiomatic expressions where ill would not fit native sensibility.
“I am sick and tired of X,” is used to mean that somebody's patience is worn out.
No native speaker would ever say ill and tired in this case.
Likewise, if someone were to drink too much and vomit, one would say, “He got sick.”
To get sick is so strongly connected with vomiting that you can even say, “He got sick on his shoes,” or “She got sick last night,” for instance and there will be no ambiguity among native speakers in the USA.
Likewise, to fall ill is never worded to fall sick .
To us, that would be just odd.
Either word might be used to describe someone’s mental illness, such as “He is sick in the head,” or “He is mentally ill,” though the phrase “mental illness” sounds right to us, and you will probably not often hear an American, at least, use the exact phrase “mental sickness.”
ell.stackexchange.com/users/7/Ryan Haber
Illness refers to a medical condition.
Sickness refers to the way one feels.
Illness often makes one feel sick, so the terms are often used interchangeably in colloquial speech.
But, one can be ill without being (feeling) sick.
Likewise, one might feel sick after, say, seeing blood, without being ill.
ell.stackexchange.com/users/69/Flimzy
A related Australianism.
Having a day off work - for the stated reason of being too ill to attend - is known as chucking a sickie .
You would never call it 'chucking an illie'.
To the question: the biggest difference is for definitions involving nausea in which case sick is used exclusively.
Hence sicking up .
Additionally, the colloquial meanings (ie not strictly related to being unwell) tend to use sick .
For example, sick and tired (see Ryan's ans.); fully sick (meaning really good ); sick to death (worse than sick and tired); you make me sick (you are disgusting, obnoxious or offensive to my sensibilities).
ell.stackexchange.com/users/89/mcalex
In Indian context both can be used interchangeably.
But there is a subtle difference between the two.
Sick could be used if someone is annoyed by one's act or behavior.
He'd be rather sick than feeling illness by the deeds of that person.
Likewise if someone has done something wrong to me, then I'd be feeling sick.
Whereas ill means that a person has been acquired by the disease.
So it might be possible that, the person might be feeling sick(feeling frustration due to suffering).
So this is the difference between the two.
But I've seen that generally people conversing in colloquial language uses the both the terms interchangeably.
ell.stackexchange.com/users/902/Sudhir
A native speaker would interpret them as having the same meaning.
You could say "I'm ill," or you could say "I'm sick".
"I'm ill" could be classed as more formal language.
ell.stackexchange.com/users/12/Liam W
Being ill refers to both long-term and short-term diseases or ailments.
It's more formal.
Being sick refers to short-term or temporary ailments, such as vomiting etc.
Besides, 'sick' is an attributive adjective , i.e., you can use 'sick' before a noun; but you cannot use 'ill' before a noun in the sense of ailment .
"The boy felt sick and went home after the third period.
"
"I've been ill with the flu for the past few days.
"
"The mother took care of her sick child.
"
ILL has some other meanings :
(1) evil/wicked : Ill men conspire against him.
(2) morally reprehensible : It's ill to keep a lady waiting.
(3) harsh/cruel : I dislike his ill manners.
SICK has the following meanings : (1) in bad taste : That is a sick joke.
(2) tired of / annoyed with :
I am sick of that song.
(3) in poor condition : My job prospects are pretty sick.
ell.stackexchange.com/users/114137/Sandip Kumar Mandal
I can say  I'm ill  or  I'm sick . But what is the difference between the usage of these terms?  I've heard that one can use  sick  for longer-term and  ill  for shorter-term, but is that really correct? How are these terms different for native speakers?
Is there any difference between being ill and sick?
ell.stackexchange.com
<word-usage><word-choice><difference><synonyms>
ell.stackexchange.com/questions/2
ell.stackexchange.com/users/3/Danubian Sailor
It is fairly common to use  ill  for longer or more serious issues, like cancer, and  sick  for more immediate things, such as nausea.  Illness is the objective diagnosis that an external impartial observer is able to make based on the constellation of symptoms which the patient presents. Being  sick  can refer to actually throwing up or vomiting, but it can also be used for being generally unwell. There is a distinction medically. Sickness often means being generally unwell and illness is an objective diagnosis.
There is a distinction medically. Illness is an objective diagnosis. Sickness is a state of being, often refers to being generally unwell.