Although this is a very old question in Java 8 you could do something like
List<String> a1 = Arrays.asList("2009-05-18", "2009-05-19", "2009-05-21");
List<String> a2 = Arrays.asList("2009-05-18", "2009-05-18", "2009-05-19", "2009-05-19", "2009-05-20", "2009-05-21","2009-05-21", "2009-05-22");
List<String> result = a2.stream().filter(elem -> !a1.contains(elem)).collect(Collectors.toList());