They can both be used to mean ill. We use 'sick' to talk about needing to take time off from work or study due to illness. For example: 'I was very ill last week.' 'I took time off work, sick, last week' 'I took sick leave last week' But more often, when we use the word sick, it means vomiting, we say 'to be sick'. For example 'She took the day off because she kept being sick' means she took time off work because whe kept vomiting.

sick and ill 疾与病的区别，你应该懂的

"Ill" is more formal. I would say: I am sick with a headache. For some reason the food made me sick (or ill). I feel sick today. I think I'll stay home from school. I am tired of that fool.....he makes me sick! (meaning I'm beginning to despise him)

I wanted to ask this question, you were ahead me. Thank you so much.

The word "ill" can also be used sarcastically. Say you have seen someone who is acting very snobbish or "rich" ( a poser) Their behavior is so obnoxious, you would say, "I think I am going to be ill" in a sarcastic way.