You get free compile time checking of valid values. Using
public static int OPTION_ONE = 0;
public static int OPTION_TWO = 1;
does not ensure
void selectOption(int option) {
...
}
will only accept 0 or 1 as a parameter value. Using an enum, that is guaranteed. Moreover, this leads to more self documenting code, because you can use code completion to see all enum values.