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Consider the following example sentence (taken from an exercise sheet):

> 熱が下がらず苦しかった。

Now consider this sentence I made:

> 熱が下がらないで苦しかった。

What is the difference?
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2014-11-25T03:41:36.573
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2014-11-26T02:33:25.147
2014-11-25T06:26:15.683
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〜ず versus 〜ないで　(and maybe also in the case of adjectives, 〜なくて）
There are several ways of saying the same thing:

> 熱が下がら **ず** 苦しかった。 - literary or stiff expression  
>  熱が下がら **なくて** 苦しかった。 - casual / conversational.  
>  熱が下がら **ないので** 苦しかった。 - using explicit ので "because".

I think ～ないで is used mostly for: ① negative imperative: "さわらないで！" (milder than
さわるな！) ② in the form of ～ないでいる and ～ないでおく: "今は言わないでおこう" ③ before various
verbs: "顔も洗わないで行っちゃった". In cases ② and ③, ないで can be substituted with ずに.

Based on these, 熱が下がら **ないで** 苦しかった and 熱が下がら **ずに** 苦しかった are both
understandable but sound slightly clumsy or unnatural, perhaps because 苦しかった
is not a verb. With a verb instead, 熱が下がらないで苦しんだ and 熱が下がらずに苦しんだ are both
fairly correct (except that 苦しんだ is rather a literary expression).
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2014-11-26T01:35:21.457
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To add to the other answer, I was taught that 〜なくて can imply a a causal
relationship, while 〜ないで doesn't.

> 電車に乗れなくて、遅刻した。 - I was late because I couldn't get on the train.  
>  シャワーを浴びないで、家を出た。 - I left the house without taking a shower. (but not
> because)

In these examples, the two are not interchangeable as far as I know.

Also, as the other answer mentioned, 〜ないでください is the standard form for
negative imperative / requests and can't be substituted by 〜ず or 〜なくて.
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