They are synonyms. "Cease fire" is usually used today, perhaps because the word "armistice" became associated with the end of World War I on November 11, and the subsequent "Armistice Day" to remember. (This holiday is now called "Veterans Day.")
I would consider a ceasefire to be potentially more temporary. Warring factions quite frequently call a ceasefire but it can last from hours to weeks. An armistice is usually more "official", with the leaders of the warring factions signing papers declaring a war to be over. I would say that an armistice might come after several ceasefires.