I count more than a dozen websites that discuss the difference between the words ill and sick. Let's add one more, although I was under the impression there is no difference, except that ill may seem slightly more polite or formal than sick to many of us.
In the United States the two words do mean the same thing, although one word may be used more than the other in certain contexts. Nobody says, "I took an ill day," for example. To some, ill suggests a more serious malady than sick. (We may speak of a "terminal illness" but probably not a "terminal sickness.") In Great Britain, however, there is more difference between the two words. To the British, the word sick suggests nausea. Sick is even used as a synonym for vomit, as in, "There is sick on the floor."
Bill Bryson, who lived in England for a number of years, says in his book Mother Tongue that the British often say ill in instances where Americans would say injured.
The reason English has so many words that mean essentially the same thing is that the language has gained words from so many different sources, and many of these duplicate existing words. In this case, sick is a Saxon word, while ill, with Scandinavian origins, was brought to England by the Normans. The Normans influenced nobles in England more than peasants, which may explain why the word ill still sounds a bit more refined than sick to our ears.