One advantage to the second method is that you can wrap it with Collections.unmodifiableMap()
to guarantee that nothing is going to update the collection later:
private static final Map<Integer, String> CONSTANT_MAP =
Collections.unmodifiableMap(new HashMap<Integer, String>() {{
put(1, "one");
put(2, "two");
}});
// later on...
CONSTANT_MAP.put(3, "three"); // going to throw an exception!