“I’m feeling under the weather”
“Jane is sick today, she can’t come to class.”
“My grandfather is ill.”
All of these sentences have similar meanings, but let’s take a look at the differences.
First off “Under the weather” is an idiom that means “sick”. It is believed to come from sailors who had to stay below decks on ships when sick (under the deck and under the weather outside).
Although it has the same meaning as “sick” since you are speaking indirectly through an idiom it is a little less strong. It is also more common to use with softeners like “a little under the weather” when you feel only partially sick.
“Sick” and “Ill” are synonyms, two words that mean the same thing, but we tend to use them a bit differently.
We more often use “sick” for short term things like a cold, flu or food poisoning that last several days.
“Ill” is more often used for long term diseases or hereditary illnesses like cancer.
“I missed work last week because I was sick.” Is common, however “I missed work last week because I was ill” is still grammatically correct and not unnatural -just less common.
“My grandfather is sick, he has cancer.” Likewise is not wrong, just a less common use of the word sick.
“Under the weather” though can only be used for short term, temporary things and would never be used for a fatal disease.