“On the contrary” vs “On the other hand”

When we use “on the contrary”, we are actually emphasizing, or reinforcing, a statement by saying it the other way round. We can use indeed to replace it.

e.g. John is not a modest person. On the contrary, he is very proud.

“Modest” and “proud” are antonyms. The two sentences mean quite the same thing.

“On the other hand” is used to connect two contrasting ideas. It is similar to “however” in meaning. 

e.g. Many people are planning to emigrate to other countries; on the other hand, they don’t want to leave their close relatives in Hong Kong.