If I'm not healthy, am I sick or am I ill?
Are these interchangeable, or do they merely overlap?
They're generally used synonymously these days, though there was originally a distinction in meaning. "Ill" generically referred to being unwell, whereas "sick" referred to vomiting — this still persists to some degree.
In modern times there has been (and still is to a degree) a distinction between upper-class and non-upper-class usage in British English. See the Wikipedia article for example. The upper-class usage, unsurprisingly, seems to preserve the more traditional meanings of the two words.
U Non-U Ill (in bed) Sick (in bed) I was sick on the boat. I was ill on the boat.
Finally, if you're curious you can take a look at the etymologies of sick and ill. The latter originally only meant "morally evil", curiously enough. This is going back to the high Middle Ages, however. Meanings have been are still in constant flux.
Be aware that sick has lately taken on some new freight: in current slang usages it is replacing awesome as the clichéd announcement of approval.
That is just sick!
That is so sick!
are not used to mean unwell or ill. And while you may not have any trouble distinguishing the meanings when applied via the demonstrative pronoun that, the word is also being applied to personal pronouns as well:
He's so sick!
can mean the speaker is commenting about some outstanding qualities the subject has. Really.