Both the words “opposite" and “contrary" are used to mean “contradictory", but their usages largely differ. The word “opposite" is used both in formal and informal English. When we talk about something that always goes into pairs, or two sections of a pair, we use “opposite". Thus black is the opposite of white, tall is the opposite of short and calm is the opposite of agitated. The phrase “diagonally opposite" is also often used in English.
“Contrary", on the other hand, is a much formal word and has specific areas of usage. The phrase “on the other hand" in the previous sentence can be replaced by “on the contrary". “Contrarily" is found in early English writings to mean the same, but it has been out of date nowadays. We also talk of “two contrary ideas".
Please note that “two contrary ideas" can differ in some sections and coalesce in others; but “two opposite ideas" are poles apart and can never have any similarity in any condition.
1. Do you know the opposite of ‘good'?
2. The pond is located at the opposite of the market.
3. Let me see the opposite (also, other) side of the page.
4. Plato could not stand poets and had no place for them in his idea of Republic. Aristotle, on the contrary, had defended them.
5. Both Ram and Rahim agreed to go to the cinema. Ram voted for a Hindi film and Rahim wanted to see an English one. Their plans are contrary, for they have the similar point of seeing a movie.
6. I wanted to watch cricket on television. My sister wanted to watch her favourite comedy show. These are two opposite ideas.