Sick and ill are words that are similar in meaning but have clear differences in usage. Sick is the less formal of the two words. It usually describes short-term diseases or ailments, like the flu, and is commonly used to refer to a feeling of nausea. Think of a sick child that stays home instead of going to school or a person who feels sick to their stomach after a roller coaster ride. Ill is more formal and is used to describe long and short-term diseases or ailments. A person who is not in good health because of something serious like cancer or pneumonia might be said to be ill. On the other hand, a person who is affected with a minor fever or cold can also be called ill. Like sick, ill also refers to a feeling of nausea. Two common combinations of ill are "ill health" and "mentally ill."  Ill is the preferred word in such cases, though sick can be used in its place. source:http://www.learnersdictionary.com/qa/sick-and-ill

There is absolutely no difference in usage or meaning. Learners sometimes try to slice and dice certain words looking for differences where none exist. This is one of those words. Additionally, there is no 'formal' or 'informal' nuance to either word. Note: This is the American English answer. I'm told that it is different in British English. { I am a native English speaker }