I don't believe there's any difference, except that "sick" is blunt and direct, wheres "ill" is polite and euphemistic. It's like the difference between "sweat" and "perspire." "Sick" isn't a taboo word, and it isn't a colloquial word. Companies will have policies about "sick days," for example, or say that "so-and-so is out sick today" or "is on sick leave." But it is a little softer and nicer to say "ill." As others have noted, "sick" is sometimes itself a polite euphemism for "vomiting." But "ill" can be used in this way, too. The word "ill" also has a range of meanings unrelated to health: evil, unsatisfactory, poor. You can't use "sick" for these meanings. For example, "The people are ill-served by the administration's present policies," or "the bad weather seemed like an ill omen." You couldn't say "sick-served" or a "sick omen." There's a proverb in English, "it's an ill wind that blows no good," which means that most bad events have at least some good in them, and it's an "ill wind"--a really really bad one--that blows no good at all. You couldn't say "it's a sick wind that blows no good."

Native speakers use these terms pretty synonymously. "Sick" is rather colloquial whereas ill is more formal but they have identical meanings in everyday conversational use.

I'm ill is more common and general and used in a medical condition I'm sick refers to actually throwing up or vomiting, but it can also be used for being generally unwell.

These words aren't synonyms in GB English. The word 'ill' is usually the only word used in Britain to refer to an unspecified illness or feeling unwell. 'Sick' has one specific meaning - 'I feel sick' means 'I feel like I am going to vomit'. If you say 'I was sick' you mean that you actually vomited. We don't usually use the word 'sick' to mean generally unwell.

Sick = cold, flu,fever... Ill = virus, rubella... You're welcome ^_^