Parts of speech are the same
Ill and sick are both adjectives, followed by be, become, feel, look, seem, to mean "ill" or "unwell"
Example 1. John is ill and can't come to school. John was sick and couldn't come to school.
Example 2. Mary looks ill. What's wrong with her？ Mary looked sick. What's wrong with her?
Example 3. Alice fell ill while on holiday. Alice fell ill while on vacation.
Example 4. If you spend too much time in the cold, you will get ill. If you stay too long in the cold, you'll get sick.
Example 5. Joe felt sick and had to go home at lunchtime. Joe felt uncomfortable and had to go home at lunchtime.
Example 6. I am sick in bed this morning, and I will report in sick. I was sick in bed this morning and had to take sick leave.
Word meaning differences
In American English, sick equals ill, but ill is more formal than sick, especially for serious, chronic diseases that endanger health. Long-term or short-term illnesses, ranging from mild fever or colds, to serious illnesses such as pneumonia or cancer, can be ill and sometimes mean "nausea", which means the same thing as sick. In British English, sick means "nausea," a feeling of stomach upset that can cause the urge to vomit, while "sick" is usually ill instead of sick.
Example 6. I was ill for a time last month, but I'm fine now. I was sick last month and now I'm better.
Example 7. She's critically ill in hospital. She was hospitalized and very ill.
Example 8. Is there a bathroom? I feel sick. Are there restrooms? I felt sick.
Example 9. If you eat that food, you'll be sick. If you eat that kind of food, you'll vomit.
Example 10. Joe ate so much that he was sick. Joe ate so much that he threw up.
Example 11. The smell there made me sick. The smell there made me nauseous.
Example 12. I can't travel by boat because I feel sick immediately. I can't travel by boat because I want to throw up because I get on the boat and get dizzy.
Example 13. Joe is very sick today. Joe was very ill today. (American English)
Example 14. He has been sick over a year. He had been ill for more than a year. (American English)
Example 15. I have to call the doctor because my daughter is as sick as a dog. I had to call the doctor because my daughter was very sick. (American English)
Rhetorical differences
Ill and sick rhetoric are different. sick is used to modify the noun (person) to mean "sick", ill can not. Sick can be used in conjunction with day, leave, and pay to form sick day sick leave, sick leave sick leave or sick pay sick pay, ill can not. Ill modifies nouns to indicate "sick" usually in conjunction with the adverb (seriously, chronically, or terminally).
Example 16. John is ill, and he is an ill a sick child. John was sick, a sick child.
Example 17. Alice was at home looking after her ill sick baby. Alice is at home caring for sick children.
Example 18. She's not at work today. She’s on sick leave. She is not at work today and is on sick leave.
Example 19. This ward is for terminally ill patients. This room is the ward for the terminally ill.
Example 20. In a clear, concise, straight forward style, they discuss the communication skills necessary to master the care of the seriously ill patient.
Example 21. This is an innovative way to improve care for chronically ill patients.
There is a difference in ill and sick's figurative rhetoric. Ill means "ominous; bad; unpleasant," and sick means "unpleasant," especially anger.
Example 22. Some believe that the owl is a bird of ill omen. Some people think that the owl is an ominous bird.
Example 23. The report will induce an ill opinion of the new immigrants. The report will cause resentment towards new immigrants.
Example 24. There may always be an ill feeling between them.
Example 25. Bill was forced to retire because of ill health. Bill was forced to retire due to ill health.
Example 26. He makes me sick the way he behaves! His behavior disgusts me!
Example 27. I am so sick of all these ads. I hate these ads.
Example 28. I got the feeling that he was getting pretty sick of me. I had a feeling that he was already bored with me.