A refugee is someone who has been displaced - by political strife, war or something pretty serious. Usually used in the mass plural (thousands or tens of thousands).
The key is that they may well wish to stay in their homeland but are unable to due to circumstance; by contrast an asylum seeker is someone actively appealing for citizenship of a foreign nation and instigating this themselves, though their underlying motives (escaping persecution, war, poverty or the political situation) are often common and they may simply be preempting the time when they would forcibly become a refugee.
The etymology of the term 'seeking asylum' is relevant to this. Courtesy of Messrs Merriam-Webster:
Etymology: Middle English, from Latin, from Greek asylon, neuter of asylos inviolable, from a- + sylon right of seizure
Date: 15th century
1 : an inviolable place of refuge and protection giving shelter to criminals and debtors : sanctuary 2 : a place of retreat and security : shelter 3 a : the protection or inviolability afforded by an asylum : refuge b : protection from arrest and extradition given especially to political refugees by a nation or by an embassy or other agency enjoying diplomatic immunity 4 : an institution for the care of the destitute or sick and especially the insane
Etymology: French réfugié, past participle of (se) réfugier to take refuge, from Middle French refugier, from Latin refugium
: one that flees ; especially : a person who flees to a foreign country or power to escape danger or persecution