There's a slight difference between the two:
An asylum seeker is an individual who is seeking international protection. In countries with individualised procedures, an asylum seeker is someone whose claim has not yet been finally decided on by the country in which he or she has submitted it. Not every asylum seeker will ultimately be recognised as a refugee, but every refugee is initially an asylum seeker.
A refugee is a person who has fled their country of origin and is unable or unwilling to return because of a well-founded fear of being persecuted because of their race, religion, nationality, membership of a particular social group or political opinion.
Just to add a little bit of detail to clarify the difference: an asylum seeker can also be a displaced person -- but not an economic migrant.