On the other hand is used for balancing contrasting points; on the contrary is used for contradicting opposite points. Check out these sentences:
- They don't want to discourage me altogether. On the other hand, they don't want to make it too easy.
- Arranged marriages are common in many Middle Eastern countries. In the West, on the other hand, they are unusual.
- We haven't kidnapped your cousin. On the contrary, we're trying to find her.
- A vulgar mind will dishonor the most noble matter by treating it in a common manner. A great and noble mind, on the contrary, will ennoble even a common matter.
Note that on the other hand, just like 'while' and 'whereas', is used to balance two facts or ideas that contrast while one the contrary, synonymous to the phrase 'quite the opposite', is used to show contradiction between ideas or facts.
- There was something I don't like about you. But, on the other hand, I'm not the least bit afraid of you.
- Tinkerbell was not all bad; or, rather, she was all bad just now, but, on the other hand, sometimes she was all good.
- Swilling the system out with liquid does not in any way accelerate or aid digestion, but, on the contrary, retards and impedes it.
- He does not demand such easy-to-wear garments. On the contrary, he will undergo extreme discomfort if it gives him a distinctive appearance.
Besides contradicting someone's ideas or suggestions, on the contary can be also used to strengthen a negative statement previously made.
- I cannot find such imperfections in the normal human body as alleged. On the contrary, the more one looks into the human body and sees it work, the better one understands it.
- He did not allow the incident to discourage her. On the contrary, he began to work twice as hard.