Ok I'm not 100% sure about this but I think a refugee is someone who has lost their home etc due to war and similar things happening, where an asylum seeker is trying to leave their country because of fear of the government, ie political asylum
A refugee is someone who seeks refuge in a foreign country because of war and violence, or out of fear of persecution "on account of race, religion, nationality, political opinion, or membership in a particular social group" (to use the terminology from U.S. law).
Until a request for refuge has been accepted, the person is referred to as an asylum seeker. Only after the recognition of the asylum seeker's protection needs, he or she is officially referred to as a refugee and enjoys refugee status, which carries certain rights and obligations according to the legislation of the receiving country.