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.).