And here's a British perspective:
In GB English, 'ill' is the opposite of 'well'. If you don't feel well and can't go to work, for example, you would say that you 'feel ill'. 'Sick' has a more specific meaning: if you 'feel sick', this means that you are nauseous and feel like you need to vomit. If you say 'I was ill', we'd understand that you weren't well, whereas if you said 'I was sick' we'd understand that you vomited.
The exception is when the adjective comes before the noun, for example 'a sick child' or 'her sick husband'. In this case, 'sick' means not well, as it does in US English. It also has this same meaning in certain phrases, such as 'a sick day' 'to be off sick' or to get a 'sick note' from the doctor.
By the way, usually when this question comes up, American members make the distinction between being sick ( a minor illness such as a cold or flu) and being ill ( a more serious long-term illness or condition). This doesn't apply in British English, but I believe this to be the case in US English.