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In "Japanese in MangaLand" Lesson 35, the "negative te form" and "the te form of the negative" were both introduced. And a table was showing the following:
simple | neg. -te | -te f. of the neg.
教える | 教えないで | 教えなくて
...
貸す | 貸さないで | 貸さなくて
But I was confused, could anyone please tell when I should use which please? Thank you!
You'll have to learn the grammar to know the differences. They are just not used in the same situations.
[好]{す}きじゃないなら、[食]{た}べなくて[も]いいよ。
If you don't like it, it's ok not to eat it.
[昨日]{きのう}は、[疲]{つか}れたから、[歯]{は}を[磨]{みが}かないで、[寝]{ね}た。
Yesterday I was tired, so I went to bed without brushing my teeth.
それを[触]{さわ}らないでよ!
Don't touch that!
OK, after further reading I kind of get at least some difference between the two.
ないで is "don't do something", while なくて can be used with いい and that means "not need to do something" or "even not do something, it is still OK".
だけど甘く見ないでね But don't underestimate me (it is not OK to under-estimate me).
郵便を出すのを忘れないでください。 Please don't forget to send the mail.
何も思い出さなくていい You don't need to remember anything (it is still OK).
...while なくて can be used with いい ← ?? 「～しなくていい」「～しないでいい」 are both fine. 両方言いますよ。 eg「思い出さなくて (も)いい」「思い出さないでいい」「心配しなくて (も)いい」「心配しないでいい」など。