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異なる
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The only verb of the bunch. To differ. と異なる: Use と when one thing is very specifically NOT like the other thing. に異なる: Use に when you want to say that something is different according to the circumstances （like xxxによって違う）
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差
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Mathematical difference. Usually used in math, or the phrase 年の差が大きい！（the age difference between them is so big!)
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食違い
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This is a BAD difference – a discrepancy or gap (i.e. a credibility gap).
$$$
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相違
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A more formal/newspaperish word for a difference between two or more things.
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間隔
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A distance or gap ;a literal space between two things:between two close-together buildings, say, or between the train and the platform.
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区別
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To MAKE a difference or make a distinction between two things. (Opposite: 混同:こんどう: to confuse one thing with a similar thing)
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距離
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Physical distance between two things:towns on a map, for instance, or the distance between an archer and a target – a much bigger distance than the related word 間隔.
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KANJIDAMAGE