WordReference Collins English Usage © 2020
‘on the contrary’
You say on the contrary when you are contradicting a statement that has just been made.
‘You’ll get tired of it.' – ‘On the contrary. I’ll enjoy it.'
You also use on the contrary when you have said that something is not the case, and you are going to say that the opposite is true.
There was nothing ugly about her dress: on the contrary, it was rather elegant.
‘on the other hand’
Don't say ‘on the contrary’ when you are going to mention a situation that contrasts with one you have just described. Don't say, for example, ‘I don’t like living in the centre of the town. On the contrary, it's useful when you want to buy something'. You say ‘I don’t like living in the centre of the town. On the other hand, it's useful when you want to buy something'.
It's certainly hard work. But, on the other hand, the salary is good.