This post follows several on logic such as here.
Contrary opposites are mutually distinguished terms. One cannot exist without the other. If one is taken away, the other is also. Examples of contraries are up and down, in and out, before and after.
More than two possibilities might also be distinguished, such as negative, positive, and zero. In that case, negative and positive are contraries for non-zero numbers.
Contradictory opposites do not necessarily exist together. One contradictory opposite may exist and the other not exist. Or if one is taken away, the other may still exist. Examples of contradictories are good and evil, true and false, right and wrong. Good can exist without evil. True can exist without false. Right can exist without wrong.
There are two errors made regarding contrary and contradictory opposites. The first error is treating contraries like contradictories. The second error is treating contradictories like contraries.
The first error is made by treating a contrary as if it could stand alone, without its contrary opposite. This is done by those who treat a contrary policy or point of view as evil. For example, one might argue that giving small states more power is better than giving large states more power, but they are contrary opinions, and it would be a mistake to say that one is good and the other evil.
The second error is made by treating contradictories as if they require one another. This is done by dualists who say good and evil are complementary principles that always exist together. Those who treat a morally wrong proposal as the contrary of a morally good proposal also make this mistake.