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.