They are exactly the same, and are used in a similar way to "however" or "on the other hand" to introduce a contrast / comparison. Put "By / In contrast" at the beginning of a sentence, with a comma after "contrast".

Example

Unemployment rose in the UK. By contrast, the number of unemployed people in Canada fell.

On the contrary

I don't think you'll need this phrase for any part of the IELTS test. It doesn't mean the same thing as "by / in contrast". We use "on the contrary" to deny that something is true, and to explain that the opposite is true.

Example

- Person 1: "You had some problems with your hotel, didn't you?"

- Person 2: "On the contrary, the hotel was great, but the airline lost my suitcase.”

TOEFL Grammar: 'by/in contrast' and 'on the contrary'

不建议同学们写托福作文的时候使用'by/in contrast' and 'on the contrary’这两个词组

理由：词海中显示But/However是强词，单词有强弱之分，这就是为什么On the countrary 在托福语法中归为递进关系而非转折

（请大家翻译下汉语：我家门前有条小河但是只能在5楼看到，地道的表达下期解析）

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