For our physical and mental condition they are interchangeable in meaning but not always in usage. People who are ill are in a sick condition of body or mind.
And people who are sick are in an ill condition of body or mind.
However, there are other usage problems. People can be ill or sick but I am sick of you (not ill of you).
And:
I am sick and tired (not ill and tired).
He was ill-advised (not sick-advised).
The ill-gotten gains (not the sick-gotten gains)