There's no difference in meaning, but there's a slight difference in formality and in regional use. 'Get' is more informal than 'become'. 'Sick' is more commonly used in US English, 'ill' more commonly used in GB English. So 'become ill' is the more neutral phrase, while 'get sick' (US) and 'get ill' (GB) are the more informal phrases. 'Sick' in GB Eng suggests nausea and vomiting, rather than general illness.

They are pretty much the same.

Background: Medical sociology has long made the distinction between illness and sickness. Illness is the objective diagnosis that an external impartial observer is able to make based on the constellation of symptoms which the patient presents. Sickness is the social role that the patient adopts as the patient and other concerned stakeholders, in relationship with the patient, interpret the meaning of the illness. -- so to answer your question, a native speaker would use both alternatively, but in the medical field, it's slightly different. source: http://ell.stackexchange.com/questions/2/is-there-any-difference-between-being-ill-and-sick