首先，二者的意思都差不多。
其次，二者不能混用，体会一下下面的例句：
Despite repeated assurances to the contrary, they have not ended their nuclear programme.
People just don't do things like that.—On the contrary, they do them all the time.
On the contrary, 要反对的是整个观点，用法类似on the other hand, 但是有订正前文所述观点的意思，on the other hand没有。也就是On the contrary后边的观点是作者真正想表达的。
to the contrary, 有指向性，反对的基本上是名词，用在句中。
至于实际使用上怎么区分？一个小技巧（不一定严谨）：
to the contrary≈as opposed to
on the contrary≈instead
