'Illegal' is synonymous with 'against the law'.
'Unlawful' often means the same as 'illegal' but it can have a broader meaning, such as 'not recognized by law'. For example, if I put up a sign stating 'Trespassers will be prosecuted' it has no legal validity (because simple trespass is not a criminal offence). Hence the sign could be described as 'unlawful', despite the fact that it's not 'illegal'.
Chris
PS: The wording on 'No Smoking' signs is determined by the provisions of the Smoke-free (Signs) Regulations 2007. They must state "No smoking. It is against the law to smoke in these premises". If an employer was to put up his own signs stating "Don't smoke. It's against the law", those signs would be unlawful. (i.e. they wouldn't meet the requirements of the law). The signs wouldn't be illegal but the employer would be acting illegally if he failed to also display the correct signs.