The formal range of meanings for each word is more or less the same, but they carry different connotations and usage. It may vary from region to region, but in the USA, it is fairly common to use ill for longer or more serious issues, like cancer, and sick for more immediate things, like the nausea involved in cancer treatment.
Additionally, sick is used in some idiomatic expressions where ill would not fit native sensibility.
“I am sick and tired of X,” is used to mean that somebody's patience is worn out. No native speaker would ever say ill and tired in this case. Likewise, if someone were to drink too much and vomit, one would say, “He got sick.” To get sick is so strongly connected with vomiting that you can even say, “He got sick on his shoes,” or “She got sick last night,” for instance and there will be no ambiguity among native speakers in the USA.
Likewise, to fall ill is never worded to fall sick. To us, that would be just odd.
Either word might be used to describe someone’s mental illness, such as “He is sick in the head,” or “He is mentally ill,” though the phrase “mental illness” sounds right to us, and you will probably not often hear an American, at least, use the exact phrase “mental sickness.”