"I'm sick" means that someone has flu' or sth similar but it's not so serious. "I'm ill" means that someone is seriously ill, f.i. cancer.

Hey Henry! They both mean the same thing, but sick can be used several way. For example: Bob is really sick because he has a flu. (health) Bob is sick for thinking all those perverted thoughts.

Hello Henryk, I see that you already have some good answers to this question, but I'll throw my own answer in there as well. As you can see even among native English speakers there is often a difference of opinion regarding our language. I can't speak for the UK, but in America 'Sick and Ill' are synonomous (they have the same meaning), in general. Except that as Aisha pointed out, we often use the term 'Sick' to describe a person who has abnormal behavior or thoughts. A brutal murderer or rapist might be called 'Sick'. Like most languages though it really depends on the context. If I call my office and say I'm not coming to work because I'm sick it is perfectly acceptable use of the word.

In British English "to be sick" means "to vomit". But in American English "Bob is sick" and "Bob is ill" are almost equal. "Ill" is somewhat more formal than "sick".