There are many complex terms and processes in the world of immigration law that it can make even the simpler distinctions seem complex. For example, consider people who receive asylum and people who are refugees.
Asylum and refugee status are two different things, even though the circumstances surrounding the individual requesting either status are remarkably similar. The main distinction between the two is that people who are granted asylum are already in the United States, and they apply for asylum through the U.S. Refugees, on the other hand, live outside of the U.S. and they file their application to the United Nations.
In either case, the individual is fleeing from persecution. This persecution could be based on many things -- religious affiliations, political opinions, and the individual's race just to name a few -- and it could lead to a person being allowed into the U.S.
But is there a way for an asylee or refugee to obtain a green card and stay in the U.S. permanently?
Yes there is. It is through Form I-485, which could grant someone who is an asylee or a refugee legal permanent residency (LPR). In order to file, the asylee or refugee must have spent at least one year in the country, and there are certainly other provisions that must be completed before attaining LPR. For example, you must pay a fee, prove your one year of residency in the U.S., supply a birth certificate or birth record, and any other necessary documentation regarding time you spent outside of the U.S. while an asylee or refugee.
However, the answer to the question is "yes." Refugees and asylees can become permanent residents of the U.S.
Source: FindLaw, "Can I Obtain a Green Card as a Refugee or Asylee?," Accessed March 31, 2015