on the contrary 恰恰相反、相反地
Use “on the contrary” when you have just said or implied that something is not true and are going to say that the opposite is true.
用來表示與先前的看法相反或極不相同，或表示先前看法的不是真的，現在說的才是真的。
E.g.
1. When I first met him, I thought he was a modest and gentle person. On the contrary, he is arrogant and ruthless.
2. A: I hear you enjoy running.
B: On the contrary, I find it to be tiresome.
by contrast, in contrast (to/with) 對照之下、與……成對比
Use “by contrast” or “in contrast (to/with)” to show that you are mentioning a very different situation from the one you have just mentioned.
In contrast 和 by contrast 意義相同，都是用來對比兩件事或兩個人之不同處。但他們的用法不盡相同。In contrast 通常會與 to 或 with 連用，用法如下:in contrast to/with+名詞。
E.g.
1. In contrast to the diligent bee, the butterfly seems to fly around without a clear purpose.
2. In contrast with most local programs, the federal government’s plan looks beyond just economic growth.
3. Cats usually sleep in the day. Dogs, by contrast, never settle down.
on the other hand 另一方面
Use “on the one hand” to introduce the first of two contrasting points, facts, or ways of looking at something. It is always followed later by on the other hand or on the other.
是對同一件事物，提供不同角度或切入點，或對剛才的說明再加入一個不同的事實。
E.g.
1. The writing class is taking up too much of my time; on the other hand, I’m learning a lot about writing.
2. I hate working in a clothing store, but on the other hand, there are also some advantages.
總結來說，在寫文章的時候，如果你已經預設了立場，要表達與之前的說法相反或強調之前的說法是錯誤的，可以用 on the contrary。
若要做客觀的觀點對比，不加入個人對錯想法，只是單純比較時就用 by contrast 或 in contrast (to/with)。
而 on the other hand 則是說明一件事情的兩面，或是不同的看法，雖有部分人士認為此片語不夠正式，但在學術語料庫出現的頻率可是這三個裡面最高的。
转自：https://www.facebook.com/ericsenglishlounge/