What is the difference between illegal and unlawful?

For all practical purposes, they are synonyms. Various sources describe possible minor differences, such as that illegal acts are criminal acts, whereas unlawful acts may be contrary to some non-criminal law, like tort law or contract law; however, if you check actual usage, I doubt whether you will find much of a pattern in that regard. The Oxford English Dictionary gives one as a synonym of the other. Unlawful obviously comes from un- and law; illegal comes from Latin in- (which means "un-") and lex ("law"). I believe unlawful is mostly just a more formal or technical synonym. Usage may very well vary in different countries, since each country has its own legal system, even though Anglo-Saxon systems are often much alike.

The New Oxford American Dictionary 3rd Edition describes illegal as "contrary to or forbidden by law, especially criminal law", and describes unlawful as "not conforming to, permitted by, or recognized by law or rules." In American English, then, illegal is used in phrases like illegal alien, where it means a person present in a country without official authorization, and which is never replaced by unlawful.

Looking at the Oxford English Dictionary, I found a note about the usage of illegal and unlawful.

Illegal and unlawful have slightly different meanings, although they are often used interchangeably. Something that is illegal is against the law, whereas an unlawful act merely contravenes the rules that apply in a particular context. Thus handball in soccer is unlawful, but it is not illegal. A third word with a similar meaning is illicit: this tends to encompass things that are forbidden or disapproved of by custom or society, as in an illicit love affair.

If something is unlawful, it means it is against the law, but not necessarily a criminal act; it can be a civil wrong, such as trademark infringement, for which the wrongdoer may be sued, but will unlikely face criminal prosecution.

Illegal describes an act that is unlawful and also a criminal act, such as drug trafficking.

EDIT: It appears these definitions aren't so cut and dry. This article discusses their usage in greater detail...

Very occasionally (C.S. Lewis?), one hears "unlawful" used in the sense of 'Moral Law', as opposed to 'man made' laws. Otherwise, for practical purposes synonymous, as already stated.

One difference to note is that unlawful is generally only used in the context of state or federal laws, wereas illegal can be used to refer to any set of rules. For example, in sports people can perform illegal moves, and when a computer program crashes it will sometimes say that it performed an illegal operation.

Within the context of the law though, they are much more synonymous save for the differences that the other answers mention.