For example, why signage in national parks says "it's unlawful to feed wildlife" instead of illegal?
In United States criminal law there is no difference between "illegal" and "unlawful." Mostly this is because of something called "substantive due process," a right which is guaranteed under the 14th Amendment and which (in part) demands that laws must not be so vague as to leave someone unsure of whether his actions are criminal. In particular, any law must enumerate which acts it is making illegal. So says the United States Supreme Court in Connally v General Construction Co. 269US385 (1926).
Since the rule on the sign prohibiting the feeding of park wildlife seems clear, perhaps the issue is whether violating the rule is tantamount to violating the law. And so it is. Step right up to 16 USC §3 - Rules and regulations of national parks, reservations, and monuments:
The Secretary of the Interior shall make and publish such rules and regulations as he may deem necessary or proper for the use and management of the parks, monuments, and reservations under the jurisdiction of the National Park Service, and any violation of any of the rules and regulations authorized by this section and sections 1, 2, and 4 of this title shall be punished by a fine of not more than $500 or imprisonment for not exceeding six months, or both,....
The principle here has a Latin description nulla poena sine lege, meaning "no punishment without the law," and is usually interpreted to mean that laws must be "praevia, scripta, certa, stricta", i.e., enacted previous to the offense, written down, certain in description, and strictly interpreted. A brief time spent in the google leads me to believe this is the way modern law works in Europe (including Great Britain), but I'm out of my depth on that subject.
There is no difference in meaning.
In this case, "unlawful" is a synonym of "illegal." The sign could just as well read "it's illegal to feed wildlife" – it would mean the same thing to the vast majority of people.
The first definition the Merriam-Webster online dictionary gives for unlawful:
not lawful : illegal
In the context of this sign and the US Park Service, there may be no difference between "unlawful" and "illegal".
Some laws prohibit certain activities, making them illegal, i.e. against the law. Most of the time there are penalties or sanctions against illegal activity.
Outside of criminal law, some laws prescribe or specify certain activities. For example, contract law specifies what constitutes a valid contract. If a contract is executed according to that specification, then it is a lawful contract. An unlawful contract fails on one or more counts in the specification. It isn't illegal, however. Just not enforceable.