The difference between asylees and refugees is largely procedural. A person who requests asylum in the United States is called an asylee. A person who requests protection while still overseas, and then is given permission to enter the U.S. as a refugee, is naturally called a refugee.
However, here is the likely source of confusion in this area. Both types of applicants must, in order to obtain their status, prove the same thing: that they qualify for protection under U.S. law, because they meet the definition of a refugee found in Section 101(a)(42)(A) of the Immigration and Nationality Act (I.N.A.) or 8 U.S.C. § 1101(a)(42)(A). That section says:
The term "refugee" means: (A) any person who is outside any country of such person's nationality or, in the case of a person having no nationality,is outside any country in which such person last habitually resided, and who is unable or unwilling to return to, and is unable or unwilling to avail himself or herself of the protection of, that country because of persecution or a well-founded fear of persecution on account of race, religion, nationality, membership in a particular social group, or political opinion . . . .
After obtaining status, the rights of asylees and refugees are similar, but not identical, as described below.
To apply for refugee status, the person must in most cases get in touch with the U.N. High Commissioner for Refugees (UNHCR). For details, see How to Prepare and Submit a Refugee Application to the U.S.
To apply for asylee status, a person who is in the U.S., and if present unlawfully has not yet been discovered and placed in removal proceedings, can submit what's called an affirmative asylum application.
If already in removal proceedings, the person can submit the application to the immigration court judge.
The least favorable alternative is to request asylum upon arrival to the United States, from the officer at the point of entry, such as a border crossing or airport. See How to Get Asylum in the U.S. for information on the application process.
Refugees have the right to remain in the U.S. indefinitely (at least until conditions in their home country return to normal). They receive a work permit, and various forms of government support during their first months in the United States. After a year of entry, they can apply for U.S. permanent resident status (a green card). Four years after that, they can apply for U.S. citizenship.
Asylees have the right to remain in the U.S. indefinitely (at least until conditions in their home country return to normal). They can apply for a work permit as soon as their asylum is approved. Some can apply for a work permit while they're still in the asylum application process, depending on timing; it's allowed where their case take a long time to process for reasons not having to do with their own requests for delay.
After a year of approval for asylee status, they can apply for U.S. permanent resident status (a green card). Four years after that, they can apply for U.S. citizenship.