Asylum refers to political protection. It is given to those who are persecuted for something in one country, and wish, no, are in need of sanctuary in a country that will not imprison, torture or otherwise perpetrate human rights violations against the individual who is at risk. Granting asylum is a measure that transcends and supersedes all international laws of immigration that might otherwise be in place.
Exile can be involuntary OR voluntary. Involuntary scenario: Emperor Napoleon (the first one, I think) was exiled by the French government to the Island of Elba.
All instances of exile are not punitive though. Here's an example of going into exile in order to avoid prosecution (not persecution), motion picture director Roman Polanski. Polanski chose exile to avoid criminal charges in the United States.
Voluntary scenario: Hypothetically, Polanski might have chosen to leave the U.S., in a self-imposed exile, to get away from the emotional anguish he associated with the U.S.A, where his 8+ months pregnant wife, Sharon Tate, was murdered by Charles Manson et. al. I mentioned that just for illustration purposes. It is well established that Polanski was a fugitive from justice and went into exile to a country that didn't have mutual extradition laws in place with the U.S.A. (And, unlike Gaddafi, Polanski was not accused of a crime as serious as mass homicide, which is a reason the country he fled to was willing to tolerate him.
Corrupt head of state example
Baby Doc Duvalier went into exile in France, once his oppressive totalitarian regime was overthrown by the people of Haiti. It was not so much a matter of France offering him political asylum, as France tolerating him to live there, an ocean away from Haiti. That's an example of exile, rather than asylum. It is involuntary, as Duvalier had to get out of Haiti. It is voluntary, in that he chose France (I guess).
Regarding @Mike's follow-up comment,
So if asylum is the act of asking for protection and exile an act of being sent away, why are we talking about "corrupt head of states going to exile in another country"... it should be "head of state asking for asylum" or "country offers asylum". Why is that?
@Mike cites a specific example, a Reuters news story about Gaddafi. However, this was NOT a situation where another country offered asylum to Gaddafi (the corrupt head of state that Mike referenced). Here is the relevant part of the Reuters article:
Burkinabe Foreign Minister Yipene Djibril Bassolet said that Gaddafi
could go into exile in his country even though it is a signatory of
the International Criminal Court, which has charged him crimes against
humanity.
"In the name of peace, I think we will take, with our partners in the
international community, whatever steps are necessary," Bassolet said,
without giving any other details.
It is important to note that Burkinabe Foreign Minister Bassolet said Gaddafi could go into exile in his country. He was willing to tolerate Gaddafi as an exile in his country only for the sake of expedience, to end civil war in Libya with associated loss of life. It was offered for the greater good of peace.
That is very different than granting political asylum to Gaddafi! It offers no guarantees of permanence unlike being granted political asylum. Being allowed entry as an exile, under conditions of duress (e.g. as Burkino Faso proposed), is an emergency measure. Gaddafi would have had to accept whatever terms he was offered by Burkino Faso, even it meant house arrest or confinement in primitive circumstances in exchange for assurance of his physical safety. The world community would have censured a country who was a signatory to the International Criminal Court, yet gave Gaddafi asylum.
This is the distinction between exile and asylum, although you may notice the words being misused, for political reasons, at times.
Bringing together a couple of good answers, the primary differences between a King and an Emperor are:
A king rules one "country" or "nation"; an emperor rules over many. This is implicit in the definition of "kingdom" vs "empire"; an empire is always made up of multiple countries that have come under full control of one governing body (typically under one man, sometimes under a small group such as a triumvirate). So, to be an emperor, you have to have an empire; typically you go from "king" to "emperor" by conquering your nearest neighbors.
A king normally rules by birthright; an emperor normally rules by conquest. Note the use of the word "normally". Kingdoms change hands between ruling families; the British Royal Family count ancestors from several formerly-competing houses, meaning that along the way several people crowned king or queen were so crowned after defeating relatives or even entirely different families (in fact perhaps the most famous line of British monarchs, the Tudors, are extinct in the male line; Edward VI, Mary I and Elizabeth I all died childless, and James VI of Scotland, Elizabeth's first cousin twice removed, ascended the throne under the House of Stuart). However, in the normal course of events, the line of succession of rule is by blood; principles of primogeniture apply more often to lines of kings than emperors. Emperors, by the nature of an empire, usually become so by conquest; you start out ruling one country, and then you invade another and replace their former system of government with one controlled by you. The line of succession of emperors can be either by blood (dynastic) or by continued conquest; the Roman Empire is a mix of succession by blood and by usurpment. The key difference is that dynastic rule is usually less important in an empire than simply whomever has the most power and influence within it as of the death or deposition of the sitting Caesar. If the previous Caesar was well-liked, then whomever he has groomed for power (whether a child or a close friend) is likely to ascend. If the previous Caesar was disliked, or didn't leave a clear line of succession, then the situation is ripe for a power grab by a previously less influential faction which then promotes its own leader into the position.
EDIT FROM COMMENT: Actually, the title of Emperor does fit with Japanese history. Prior to the 11th century, Japan was a collection of feudal states basically ruled by the landowners. The Emperor was appointed by these aristocrats to resolve disputes and provide a "unified voice" over these many feudal lands. In the early 11th century, the Kamakura shogunate was formed in the aftermath of the Genpei War, which took governmental power from the aristocrats and Emperor and placed it with the military akin to a junta. During this time, and the following Ashikaga and Tokugawa shogunates, the feudal states on the islands of Japan became unified into what we now know as a single country, and the Emperor's position evolved into more of a spiritual leader akin to the Pope. The importance of bloodline was always central (which does indicate a king), but as the Emperor became the head of the Shinto faith the bloodline came to signify the Emperor's status as a descendant of Amaterasu, and thus having divine legacy but not necessarily the mandate to govern.
In the late 1860s the Tokugawa shogunate fell and power to govern was restored to the Emperor (at the time Emperor Meiji, hence the "Meiji Restoration"). From this time until the end of WWII, Japan did indeed become a true empire, extending beyond the home islands to assert control over Manchuria, most of Southeast Asia and the Indonesian archipelago. The treaty that ended WWII allowed the Japanese people to retain their Emperor as a cultural icon, but stripped the office of all power to govern. The position and its trappings are defined in the current constitution as "a symbol of the Japanese state" akin to the British Crown, but while the Crown retains some key sovereign powers like command of the armed forces, the power to veto and to dissolve Parliament, etc, the Japanese Emperor performs mainly ceremonial and diplomatic duties and has no sovereign power whatsoever.
Best Answer
Integrity is about conduct; honesty is about adherence to the facts. The person who without fail submits his timesheet every week, seeking clarification if unsure how to charge some time, is acting with integrity. He is probably also acting with honesty, ensuring that the numbers he reports are correct.