Word Distinction: Contrasts

On the contrary is used to show complete disagreement with what has just been said.

"Does it rain a lot in the desert?" "On the contrary, it hardly ever rains."

On the other hand is used when adding a new and different fact to a statement.

It rarely rains in the desert, but on the other hand it rains a lot in the coastal areas.

In contrast is used to show the (surprising) difference between two very different facts:

It is hot in the desert in the day, but in contrast it's very cold at night.