As usual, DOBJG has a good discussion of ~nai de and nakute. Some extracted points:
* If a nai de clause expresses something one is expected to do, the translation is 'without ~ing', but if not, it is 'do not do ~ and ~', as in Ex. (a).
(a) 中田さん は 大阪 に 行かないで 京都 に 行った。
Mr. Nakada didn't go to Osaka; he went to Kyoto.
* ~nai de is often used to combine two sentences but can also be used with kudasai, hoshii, morau, and oku.
(c) まだ帰らないでください。
Please don't go home yet.
[more examples given in text]
* ~nai de is used only with verbs. It cannot be used with adjectives.
* ~nai de can be replaced by nakute only when the nai de clause indicates some cause for human emotion which is expressed in the main clause. Otherwise, the nakute version is unacceptable. [examples given in text]