Am I correct in thinking the difference between these is that ないで is more a neutral descriptive and なくて is used if there is a reason attached?
Is なくて used for sequencing things, a negative in this case, such as the -て form is used when stringing together verbs and adjectives in general?
Any correction or addition welcome, as previously I’ve dealt mainly with -ないで as the negative -て form and I want to be completely clear in how and why they’re different!
Hey @marlowe
It is exactly like you wrote, I have written longer explanation so you will understand all the nuances.
ないで can be used only with verbs. なくて, on the other hand, can be used with nouns, adjectives and verbs.
But if なくて is used with a verb (not in fixed expression), just like you wrote it has to express reason or cause. (ないで doesn’t have this requirement, also when なくて is used with nouns and adjectives it simply means “not, and/ is not…, but”)
**In other words, if you simply want to say: “not and” for verbs your default choice is ないで and for adjectives and nouns なくて。If you want to state a reason/cause then you can use なくて. (though it is usually used in certain fixed contexts, more here, in other cases から and ので are better)
In addition to that: ないで and なくて have their own set expressions in which they sound natural:
ないで is the only one that is used with auxiliaries like ください、ほしい、おく。
それをしないでください、
Plesae, do not do that.
それを言わないでほしい。
I do not want you to say that.
パスタはあるから買わないでおいた。
Because I have pasta, I didn’t buy it (in preparation for something).
なくて is preferred in:
Verbなくて+ すみません (though ないですみません is more or less acceptable)
なくてもいい - don’t have to (like above, ないでもいい is also more or less acceptable instead)
忙いそがしいなら、行いか なくてもいい です。
If you are busy, then you don’t have to go.
Or なくては(いけない/ならない) (cannot be rephrased to ないで)
朝ごはんを食べなくてはいけない
You have to eat breakfast.
There is also another expression, ずに, which is formal version of ないで (and like ないで cannot be used with nouns and adjectives).
Cheers 
That’s great, thanks very much - I’ve heard of ず as a negative auxiliary verb, but haven’t come to it yet on bunpro, so look forward to having it properly in my linguistic arsenal :'D
Thank you so much for this explanation, I was about to ask a very similar question and discourse helpfully recommended this post.
Honestly it might be worth adding a little paragraph contrasting ないで and なくて in the なくて - Japanese Grammar Explained | Bunpro entry. At least it would have been useful for me.
Admittedly I just noticed that there’s this very helpful briefjapanese post that you wrote linked in the resources but I admit that I missed it the first time around…