In almost all cases, they mean the same thing and are used in the same way. However, "sick" is more common and perhaps less formal. "He's ill" sounds a little pretentious. Also, "ill" is part of some compound words: "ill-mannered", "ill-prepared", and such. "Sick-mannered" doesn't work--in those compounds, only "ill" works. Colloquially, "sick" has some special uses. "That's sick" can mean "that's cool." "He's sick" can also mean "he is depraved/disgusting" rather than literally "he is ill." It's a bit confusing, I know. But in most circumstances, when you are actually talking about a sickness, they're interchangeable.

When something is great, amazing or cool, a certain generation will call it "sick". The generation after them will call it "ill".

Sick cannot be replaced with ill in the following expressions:- sick (n) = vomit to be sick (v) = to vomit to be sick of (v) = to be fed up with sick (n) = mentally ill sick (n) = crazy but cool (this is kid's slang, and I do not advise its usage)