None. But 'unlawful' and 'unlawfully' appear in the text of statutes, Acts of Parliament. 'Illegal' and 'illegally' rarely, if ever, do: "Whosoever shall unlawfully and maliciously..." and not "Whosoever shall illegally...". There's no clear reason for this practice. An act which contravenes the statute is an illegal act which the relevant section in the statute may have defined with the word 'unlawful' : "It shall be unlawful for any person to..." ! Either way, it's illegal or unlawful.
A difference could be conjured up, that an illegal act is one that contravenes a statute, a law, and an unlawful one is one that contravenes common law, the law made by judges over the centuries and not made into a statute, but that would be mere ingenuity . We would say that murder was always ' illegal' even though there wasn't a statute saying so and, indeed, anything other than the traditional definition of it, judge made, which was only slightly amended by statute in recent years.