Hi Gina,
As far as i know that those words hasnt that different. You can use both of them as you want because they have the same different. From my limit knowlegFrom my limited knowledge all what I know that the only different between those 2 words. That one of them is British and the other is American. That’s the entire story.
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However, mainly people are using the word sick rather than ill. For example you can say I feel sick, and you can say I feel ill. But manly you won’t hear people saying that as I know. There is no fault or correct one its all that I feel sick sound more nice than I feel ill
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On the other hand you can say I was ill yesterday. It sound more normal than saying I feel ill. But for me I wont tell you that I will ever say I feel ill. It just like that.
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That’s all I hope that I have helped you a bit, wish you all the best.
Best regards,
Ayda