Asylum is available to anyone in the United States, regardless of status, who has been persecuted or has a well-founded fear of persecution in his or her home country based on race, religion, nationality, membership in a particular social group or political opinion. It is also possible to apply for withholding of removal based on the Convention Against Torture.
Naturalization refers to the process for obtaining US citizenship. Anyone born in the United States is automatically a US citizen (except children of foreign diplomats who become permanent residents), and some children born abroad may be citizens by virtue of "derivation" from a US citizen parent or parents.