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.
A quote from the MW Dictionary of Synonyms (without examples here):
"Severe is applicable to persons and their looks, acts, thoughts, and utterances or to things (as laws, penalties, judgments, and styles) for which persons are responsible. In all these applications it implies rigorous standards of what is just, right, ethical, beautiful, or acceptable and unsparing or exacting adherence to them; it not only excludes every hint of laxity or indulgence but often suggests a preference for what is hard, plain, or meager (a severe teacher). Very often the word suggests harshness or even cruelty (severe criticism)."
"Harsh suggests a nature that is unfeeling, cruel, and indifferent to the pain it inflicts (a harsh critic) or when applied to things, effectiveness in promoting discomforts or in imposing rigors (a harsh rebuke)."
Compare the following examples (taken from different sources):
The country has come under severe criticism for its human rights record. [strong, neither bad not good]
Many people feel the punishment should have been more severe.
It may seem harsh to punish him, but he has to learn that this kind of behaviour is unacceptable.
He later regretted his harsh words. [negative]
He accused her of being unduly harsh. [negative]
Nothing can justify such harsh treatment of prisoners. [negative]
Best Answer
Notice that in your own examples, you vouched for a person, but guaranteed an action. This is exactly how I'd use these words, and it does seem to reinforce your intuition: "vouch" is about a person's character; "guarantee" is about actions or things. We might also say, however, that "guarantee" emphasizes the future, while "vouch" emphasizes the past. To explain my position, I'll call on the Oxford English Dictionary.
Guarantee
The OED lists the following meanings for the verb "guarantee":
All of these definitions and examples state that something will happen in the future, or promise to do something or cause something to happen. In the first example, "Written languages guarantee a systematic pronunciation," the claim as I understand it is not that written language can't exist unless a systematic pronunciation is already in place; the claim is that written language tends to create a systematic pronunciation because we come to associate certain symbols with certain sounds. (Obviously, this claim would not apply to all writing systems.)
"Guarantee" is also often applied to products ("We guarantee this vacuum cleaner for life"); again, this is a statement about the future: the vacuum cleaner will not break down, and if it does, the manufacturer will replace it.
Vouch
"Vouch" has quite a few meanings, many of them archaic and/or specific to the legal realm. "Vouch for", however, has only these three:
I take it that the first meaning is the one you had in mind. My belief that "vouch" emphasizes the past, rather than the present, rests on the fact that even though you vouch for a person with an eye to the future--securing a job for her, for example--you're vouching for certain qualities, like character, experience, expertise, or personality, which are already possessed by that person. You're not guaranteeing that you'll take over the job if she fails; you're just saying that she has the expertise and background necessary to succeed at it.